Why do electrons absorb and re-emit photons? Up to a certain time, I was told photons a.k.a. light was just a wave of energy. Then I was told, no, light is actually a particle. And electrons in an atom absorb and re-emit it. But why do electrons bother to absorb and re-emit light and not just let it pass all the time? (An electron would also be unstable by absorbing the energy and thus it re-emits it but in the first place why does it absorb it?)
*Note:- A similar question was asked earlier (How does an electron absorb or emit light?) but my question is not the same. The earlier asked question was how does it happen and I ask why does it happen.
 A: 
And electrons in an atom absorb and re-emit it. But why do electrons bother to absorb and re-emit light and not just let it pass all the time? 

There is a basic misunderstanding in your question.
An electron is an elementary particle of fixed mass. It can scatter off a photon, (which is also an elementary particle); if accelerated it can emit a photon, but it does not absorb it, because the electron's mass is fixed, and if it were able to absorb a photon - at the electron's center of mass - the mass would have to change, which contradicts observations and special relativity for elementary particles.
The terms absorption and absorbs are not usable with free electrons. It is the bound electrons in an atomic system, which may change energy levels in the atom when the atom absorbs a photon. So it is not the electron that absorbs the photon, but the atom.
The atom has energy levels, and if the photon energy coincides (within a small  $ΔE$, the width of the energy level) with the transition energy of kicking an electron to an empty energy level, then the atom can absorb the photon (not the electron). So the answer to "why", above, is "because the photon has the appropriate energy to transfer the electron to an empty energy level".
If the photon energy does not coincide with transition energy of the atom, the photon may scatter with the spillover electric fields of the atom or molecule either elastically, or transferring energy and a lower energy photon continues on its way.
The relevant thought to keep is that an elementary particle cannot absorb a photon. Composite ones as atoms, molecules and lattices, can.
A: Imagine a cubic box at thermal equilibrium with a room a temperature $T$ and let's think that in this box there is an electromagnetic field (i.e. photons) and also a gas of atoms (i.e. fermions).
As you probably know, the photons are continuously absorbed and emitted by the walls of the box and they tend to reach the Planck frequency distribution at the thermal equilibrium. It is important to notice that this process of continuous absorption and re-emission of the photons by the walls of the box (i.e. by the matter!) is always present when you put together matter and light. This is fundamental if you want to reach the Planck distribution, because the latter has got chemical-potential $\mu=0$ (i.e. the energy cost in order to produce (or killing) a photon is practically zero).
If this framework is clear, now you have to for sure understand that these photons are moving into this box. During the motion they will scatter with the electrons of the atoms of the gas because the cross section elctron-photon is not zero: this scattering process characterize the interaction between photons and electrons and so, the result excitation of the atoms.
The answer of your question can be:
They interact with the electrons, because there is a continuous re-equilibrium process of light and matter that live together in order to reach the thermal equilibrium of photons given by the Planck distribution. This process is made of absorption and emission of photons by the matter. It means that photons are moving, but if they are moving it means that there is a non-zero scattering-probability -> interaction.
A: 
But why do electrons bother to absorb and re-emit light and not just
  let it pass all the time? (An electron would also be unstable by
  absorbing the energy and thus it re-emits it but in the first place
  why does it absorb it?)

A similar question could be asked about macro objects, say, a pendulum.
If you push a pendulum it is going to up and then it goes down. So, why, you could ask, does it bother to go up, if it is going down afterwards? Why does it absorb the energy of a push instead of just ignoring it?
I guess a simplistic answer is that it absorbs the energy because it gets a direct hit and it's not up to the pendulum to decide whether it should take it or just ignore it.
A: It's really down to two questions: why do electrons interact with photons, and why do atoms absorb photons?
Why interact with photons?
One can understand why electrons interact with photons by considering relativistic quantum field theory. In order to combine quantum mechanics with special relativity, you have to think of reality as consisting of "quantum fields". A field is something that has a value at every location, for example $\Phi(x,t)$ might be a (time-dependent) field, the value of the function signifying the value at every point in space (and every time t). A classical, non-quantum, field simply has a value at every location - you can think of it as the height of some system, say the deviation from equilibrium of an oscillator, at every point in space. A quantum field instead has a quantum system at every point in space; you can think of it as having a quantum harmonic oscillator at every point in space. The state of the point-like system, i.e. the deviation of this oscillator from equilibrium, is the "height" of the field at that point in space.
Now a core principle of quantum mechanics is that the phase of the quantum state does not matter. In order to carry this principle into a quantum field, the equations describing the physics of the system, known as the Lagrangian, has to not change if we change the phases of the states of the points in space. This requirement is known as "gauge symmetry". Now it so happens that it's rather difficult to build a gauge-symmetric Lagrangian using only standard expressions like derivatives. Instead, in order to maintain gauge-symmetry one has to introduce another quantum field, known as the gauge-field. This is the only way to maintain gauge symmetry, i.e. to maintain the requirement that the phase of a quantum state has no physical meaning.
So if you try to build laws of physics (a Lagrangian) to describe a simple matter field (e.g. an electron's field), you need to introduce an additional "gauge" field that interacts with it. The waves in the matter field will be the matter particles, such as electrons. The waves in the gauge field will be force-carrying particles, such as photons.
To summarize then, the reason an electron interacts with photons is that an electron is really a wave in a quantum (relativistic) field, and these waves have to interact with waves in the (gauge) electromagnetic field, which we call photons, in order for the electron's field to be a quantum field (i.e. for the phase of the point-like states to lack any physical meaning).
Why do atoms absorb photons?
Anna v beautifully explained already why an elementary electron cannot absorb a photon - it has to scatter it instead, as the electron's energy and hence mass cannot increase in its rest frame. But why is it that atoms absorb photons?
The important point here is that you cannot turn the electromagnetic interaction "off" for one effect while keeping in "on" for another. If you build an equation describing an electron that's attracted to a positive nucleus by the electromagnetic force, then this same system will also be affected by waves in the electromagnetic field.
So the same equations that describe the stable orbits (the electron levels/orbitals) due to the electromagnetic interaction with the potential energy of the nucleus, also describe a response to an electromagnetic wave (usually dealt with only as a perturbation off the stable state). And this interaction with the waves amounts to annihilating a normal-mode of the wave (annihilating a photon), while at the same time increasing in energy to maintain energy conservation. (Or conversely creating a normal-mode wave while dropping in energy.)
A: I think this is a very good question - in the sense that it is actually hard to answer in a clear and intuitive way. Here's my take, and it is admittedly what Terry Pratchett used to call a good lie (that is, it gives the feeling of being intuitively 'right', but probably isn't factually true):
In physics, we have fields, which are described as something that permeates space; the electric field is one such thing. On the other hand, we have 'masses' or 'charges': some quantity that intereacts with fields of given types; the electric charge interacts with the electric field, for example. What this means is that a charge somehow surrounds itself with a field, and if you change the position of the charge, the field will tend to move with it - but this doesn't happen instantly, the change to the field propagates out from the charge following a wave form. I'm trying to pick my words carefully, that's why it sounds a bit woolly - the way the field changes can be described using sinuses and cosinuses, basically.
This is similar to what happens with a cork floating on water: if you push the cork, the water wobbles, a wave spreads out, then fades away - this is a 'wave particle' if you like, only it spreads evenly in all directions - I don't think we actually know why photons seem to be more particle like than this. However, this example also illustrates how the field can influence the particle: when a wave passes the cork, it wobbles, and the same no doubt happens to electrons.
Some of the previous answers have postulated that free electrons don't absorbe photons, and I feel that is not entirely true: if a photon hits an electron, it will lose some energy, which is transferred to the electron; in my view that means some of the energy was absorbed. We have never seen a photon being fully absorbed by a free electron - and it doesn't fit with current theories, but I don't think the state of experimental physics is able to rule out that it can occur. Also, I don't feel convinced that we can explain why and how a photon can be fully absorbed by a bound electron; quantum mechanics feels rather sketchy on the details - this is perhaps not surprising, since it is a probabilistic/statistical theory: it describes what happens on average, not what necessarily happens to individual particles.
Sorry, if that last part was a bit ranty: I'm a mathematician trying to make sense of the fundamentals of physics, and having a hard time.
A: The question is "why do electrons bother to absorb and re-emit light and not just let it pass all the time? "
Nearly all of the time atoms do not absorb and reemit light. This only happens for photons that are resonant with an excitation frequency of the atom. Non-resonant photons are not absorbed and certainly not reemitted. At very high temperatures, such as in a plasma when thermal radiation occurs close in frequency to atomic excitation energies, the conditions can be right for absorption and reemission.
Also, absorption and emission are not the only ways by which light interacts with matter. Light can be scattered by molecules, elastically by Rayleigh scattering and inelastically by Compton scattering. In dielectrics such as window glass reflection and transmission occur by the mixing of light with electronic excitations of the dielectric. Some of this light is also absorbed by inelastic scattering with such excitations.
In general no relativistic field theory is required unless very accurate atomic energy levels are required, heavy atoms are involved or esoteric effects such as vacuum polarisation kick in.
A: Absorption and emission is how we describe the interaction between electrons and the electromagnetic field using quantum field theory. If the photons didn’t scatter off electrons they would not interact.
Basically if you couple light fields and matter fields and quantize you must get a process where the quanta of the fields (electrons and photons) must scatter (absorb and re-emit).
