Why are there only four fundamental interactions of nature? Is there an answer to the question why there are only four fundamental interactions of nature?
 A: The answer "because we do not need more" by @rubenvb is fine.
Studying physics, you must realize that physics is not answering fundamental "why" questions. Physics uses mathematical tools to model measurements and these models have to fit new data, i.e. be predictive. As long as the models are not falsified, they are considered valid and useful. Once falsified, modifications or even drastic new models are sought. A prime example, quantum mechanics, when classical mechanics was invalidated: black body radiation, photoelectric effect and atomic spectra falsified efforts of classical modelling.
Physics using the appropriate models show "how" one goes from data to predictions for new experimental data. Looking for "why" in the models, one goes up or down the mathematics and arrives at the answer "because that is what has been measured"
A: Because we don't need more.
Well, we haven't found any evidence of any others. And until then, there's no need. Granted, some experiments might show indication of something else going on that pushes revision of the Standard Model.
On the mathematical side, this can be explained from symmetry: the Standard Model Lagrangian obeys a certain set of symmetry operations, which physicists assume to be valid. From this Lagrangian, using the Quantum Field Theory formalism, the separate "fundamental interactions" can be derived. From Wikipedia (emphasis mine):

The global Poincaré symmetry is postulated for all relativistic quantum field theories. It consists of the familiar translational symmetry, rotational symmetry and the inertial reference frame invariance central to the theory of special relativity. The local SU(3)×SU(2)×U(1) gauge symmetry is an internal symmetry that essentially defines the Standard Model. Roughly, the three factors of the gauge symmetry give rise to the three fundamental interactions. The fields fall into different representations of the various symmetry groups of the Standard Model (see table).

The three interactions mentioned here are slightly different categorizations of your four fundamental interactions, but essentially the same (except maybe the Brout-Englert-Higgs part).
The fourth, gravitation, is assumed to be somewhat related to the Higgs part of the Standard Model, and doesn't yet quite fit in the rest of the Standard Model.
A: Taking a slightly different angle on the question the answer that "we don't need more" is definitely correct, but further, I would say we don't need more, and the less the better.
To answer why this is we need to know about a problem sometimes referred to as overfitting. The idea behind overfitting is this; say I design a predictive model, for example I wish to obtain a relationship between the qualities of oak sap and the age of an oak. I learn the ages of a 100 oaks from records of their planting and I measure the viscosity of their sap, the colour, the density, the water content, and 46 more observables. I start by just looking at the relationship between viscosity and age. Maybe using a quadratic model, with 3 parameters. It's quite a strong relationship, and I can get within about 20 years of the age of each tree using just the viscosity of the sap. But there are 49 more variables! So I construct a complex model for the age of a tree that predicts how the age should effect all 50 variables, and then I fit the 150 parameters of this model. By tuning the complex model I can predict the ages of all 100 trees to within a day, how wonderful. Then someone shows me another oak, they say "I know how old this oak is, let's use it to test your model". The simple, viscosity only, model an the complex model give very different answers. The simple model is much more accurate at predicting the age of the new tree. Why? Because the complex model was not really learning the ages of the 100 trees at all, it was learning to identify the 100 trees using noise in the data that was unrelated to their age, and then I had tuned it's parameters so that it would give me the correct weight for each tree. It wasn't possible to do this for the simple model, because there were not enough  parameters available for me to tune, so the simple model was forced to find a genuine correlation if there was one to be found. The complex model is overfitting.
And so the more complexity there is in any model the greater it's tendency to overfit. When a model starts overfitting it loses predictive power. I am quite confident that someone with enough time an patience could derive a model that had 5 forces in it. They might even find a model that was more intuitive, or computationally easier to work with. But that more complex model would be less likely to fit new unseen results, because its complexity would be tuned to the results that we have seen, and so use the noise to identify the data and give us the answer we tried to fit. This noise is not something we wanted to fit, and a simpler model is less likely to have the capacity to fit noise in the first place.
A: There's no reason to believe that there should only be four. Four is the highest number of fundamental interactions that we've seen so far, but there may be other, extremely weak* forces that we simply haven't observed yet. There are plenty of experimentalists working on looking for new macroscopic fundamental forces even today; in fact, as an undergrad I spent most of my research doing exactly that!
There's also plenty of reason to believe that there could be more than four fundamental forces. The main paper that we used to justify our particular brand of research derived 16 exchange potentials from nonrelativistic QFT, only one of which actually matched the familiar scalar $1/r$ potential common to gravity and electromagnetism (https://arxiv.org/pdf/hep-ph/0605342.pdf). There are many, many other papers that predict the existence of additional fundamental forces in different ways, so this is a fertile experimental research field.
*Well, either extremely weak, or otherwise occupying obscure corners of phase space (like a force that is strongest at scales of a micron, or one that requires a particular combination of spin polarization and translational velocity that is not often tested in experiment).
A: The search for a Unified Field Theory posits that there is one fundamental field. It is expected that all four forces will be unified at the Planck scale, which is many orders of magnitude outside current experimental reach, although gravitational observatories and cosmological models may provide insights. 
Three of the four fundamental forces have been integrated with each other, and mass and special relativity, in quantum field theory (QFT). That only leaves gravity, which the leading attempts to unify are string theory and loop quantum gravity. It should be noted that this will have to involve dealing with the nature and emergence of time, which is not included in QFT. 
The three linked fundamental forces have been shown to arise from three fundamental symmetries and their associated conservation principles, in time (energy), in space is translation and rotation (momentum), and internal symmetry (particle identity and quantum numbers). These considerations give rise to understanding particle-families as local gauge symmetry symmetry groups, that pair boson and virtual particle exchanges into what we call the fundamental forces or fields. We have the Higgs boson from which mass derives, but no evidence for the graviton expected to mediate it. 
So, the fundamental forces are revealed as being related to the properties of our universe's topological properties. String theory posits a wider domain of possibilities for physical laws. We find that significant properties giving rise to complexity in our universe seem to be a result of fine tuning, and we live in a fine tuned universe which appears to violate assumptions we make about 'naturalness'. It seems inevitable that there is some kind of probability space in which these fundamental constants are different, which is typically called the multiverse, making the fine tuning problem one of locating our own universe within that. 
The broad responses to these issues are, disputing the fine tuning (in particular as subjective anthropocentrism), criticism by lack of imagination (our inability to explore 11 dimensional topologies gives us very little idea of the other options), and the anthropic principle (many variants, including the simulation hypothesis). 
I do not claim to be an expert in this, or even very knowledgeable. It only seemed to me this ground was not covered in other answers. From which I see, a high degree of skepticism and reluctance about these issues, despite them being self-evident results and open problems of modern physics. I would love anyone to correct me with references and where possible by disputing statements in my references. 
A: I think it is wrong to say that "why" questions are beyond the purview of physics, or that there can never be an answer to this question. Very often physics can explain why something is true in terms of more fundamental principles. Sometimes the why of those more fundamental principles can be explained in terms of principles that are even more fundamental. But there is always a lowest currently understood layer, about which all one can say is "this is the model that fits our observations best."
For instance, when Kepler worked out his orbital laws they were just basic facts of nature. If you asked why they did that, the answer would be that "it is what has been measured," to quote anna v's answer. 
But then Newton found a reason why Kepler's laws were true: The principle of universal gravitation. And now we can say that the inverse-square law of gravitational attraction is approximately true because of the way mass causes spacetime to curve. But at that point we must stop. We know of no deeper reason why mass should do that to spacetime. It is just what we observe.
This is the state of affairs with the four forces. The best models we have include them, and that is the only reason we can give now. But there is no guarantee that we will not discover some deeper theory that can explain the existence of four forces, rather than three or five, as a consequence of some more fundamental principle. But in some sense we would then be back where we started, because that principle would not have a reason why it is true.
A: Because that's the smallest number of laws we've been able to figure out to explain what we observe as well as possible.
Alternative: Physicists have been working for more than a century to reduce the number of fundamental laws, preferably to one, and so far, they've gotten it down to four.
A: 
Is there an answer to the question why there are only four fundamental interactions of nature?

No, and there never will be one
We have no science at all which can answer any meaningful "why" question about the structure of the universe, especially not about why things we do not observe are missing. 
The best we can come up with is to observe a bunch of features and figure out somewhat fitting mathematical/physical theories about how they might be interrelated. And we get that wrong all the time.
My word "meaningful" up there means going down to the "lowest turtle" when explaining something; i.e. not simply an answer which stems from theories we have which seem to fit our current understanding of nature.
For example: we know that nothing can go faster than c mainly because according to Einstein, it would take an infinite amount of energy to get something with mass even up to c, not to mention faster. But this is, in the sense of this answer, not an answer to the question of "why can nothing go faster than c?". I interpret that question as "Why is our universe structured in a way that you need infinite amounts of energy to get to c?".
In fact, on any given day someone might find an error in our theories and figure out a way to move faster than c, or do time travel, or find a 5th fundamental interaction. As we cannot even know anything about the universe with absolute, fundamental security, we can certainly not answer a "why" answer.
A very nice video about this: Richard Feynman. Why.

So I am not going to be able to give you an answer to "why magnets attract each other?" except to tell you that they do, and to tell you that that's one of the [...] different kinds of forces [...] [...and to give an explanation in more difficult terms...]

This is not exactly the same kind of thing we're talking about here, but should show the spirit when answering "why" questions. 
Outlook
This is a fascinating topic, and you could do worse than checking out the antrophic principle and the moat of philosophical topics hidden behind that. This principle comes in two versions:


*

*The strong antrophic principle basically says (my wording) that the universe was tuned "just so" to enable intelligent life (us!), i.e. with intent.

*The weak version says "if the universe were not able to sustain life like ours, we would not be here to ask the question".


The strong version veers right into the direction of beliefs and religion, and while for many people it may be valid, it certainly is not really applicable here on Physics.SE (and while there may be believers amongst physicists, it certainly does not give us any new information).
The weak version is obviously true, and it leads to a bias about what features our universe must have. But it also does not answer any kind of deep, substantial "why" question.
A: Physics (and science generally) aims to answer the question of "What rules does this seem to follow". 
We are human,so we hope to get deeper insights, and questions like "why?" inspire us powerfully, and we do ask these questions a lot. But the hard science research focuses on observing and testing rules that things seem to follow, even if as humans we are guided by wider ideals, and we imbue their findings with deeper meaning. 
So in physics we ask what rules the physical universe seems to follow; in biology we ask what physical rules living systems (including connected systems such as ecospheres) seem to follow, and so on.
To take an analogy, there is a child's game where one person picks out (say) specific cars on the road, and the other person has to ask questions to guess what rule is being used. It's a bit like that.
What this means is that the idea of 4 fundamental forces is not how the universe works. Well, it might be, but it probably isn't, and philosophically its different anyhow:


*

*The universe might work on any or no principles, including ones we can't actually conceive, or ones we would consider irrational. If that's how it works then that's how it works and we are stuck with it. (Of course it may not, but the point is, if it did then we couldn't handwave it away)

*We have limited knowledge (basically whatever we know, test, or find at this time) so we are limited by our toolbox as to what we can conclude. If our methods don't yet include the necessary tools, we can't get the right answers. 

*A model is not the same as reality. At best it says "This best approximates and predicts what is going on, and provides some logic for how it might be". But it's intimately tied to the observer, and it's never the reality itself.


So right now as of 2018 the best "what rules does it seem to follow" that we have come up with, has 4 forces in it. Tomorrow's best might have 5, or 15, or 1, or not have any notion of "fundamental forces" at all, and the day after's might be different again.
And that,really, is why we say there are 4 fundamental forces. Not because there are, but because our current best guess of "what rules does it seem to follow" is a set of rules based on a concept of quantum fields/interactions/forces and includes 4 of them. We test it hard in all kinds of ways, and it seems to really reflect the rules that the universe follows. Tomorrows new observations and new best guesses - who can tell?
A: There are no theories which can predict the number of fundamental interactions in Nature, and they never will exist. Throughout time, Physics has been dominating the fields of larger energy scales. At ancient Greek times, or even probably before, natural philosophers realized about Gravity, Electricity and Magnetism, as their effects are noticeable at the energy scales we humans can have sensorial experiences. When trying to dig deeper into matter's structure (lower spatial scales=higher energy scales), physicists realized there were another two types of interactions that we cannot notice them by a daily experience as falling from the bicycle or rug with the coat or playing with magnets.
Maybe digging deeper we could find another type of interaction, maybe not, so the answer to this question is a bit redundant. I have seen other answers like "why we need more" etc. If we just stick to this idea, then we are being lazy minded and with an enormous lack of enthusiasm, as physics through years has been renewing itself according to our ways to approach the laws of Nature.
