How Statistical Physics? It's a common fact that in physics, we use statistics (or maybe probabilities ) to describe the behaviour of a system. It was from the statistical analysis of a system where quantum statistics arose and then the theory of quantum mechanics began.
How is possible to make such descriptions and to construct theories fully capable of predicting and describing the (probabilistic) functionality of macroscopic system, quantum or classic or to be more specific:
What are the attributes that hold for a system with many particles that allows us to study their hole behaviour as a system, with statistical physics?
 A: We can use statistics by being willing to ask different questions.
No individual particle has a pressure or a temperature. But we can ask about subsystems with particular pressures or temperatures.
So we group collections into subsystems that are large enough and regular enough to have collective properties like pressure and temperature.
Then we can use statistical methods to find out how constraints on the whole subsystem (such as volume or total energy) can affect the collective properties like temperature or pressure.
This is a bit like cheating. We decide to only focus on things like temperature and pressure that only exist for large subsystems and then we see what kinds of constraints on the large subsystem such as total energy can affect it. And the real answer is that the total energy alone does not determine exactly what happens but we can set it up so that for a large enough system it usually acts in a certain way. So then we focus only on the usual behaviour.
Like all science we study it because it works and is worthwhile. Besides, it's not like we'd actually be aware of the exact state of a subsystem if the subsystem had a truly huge number of components anyway.
A: Many-body systems and other complex phenomena exhibit what is known as emergence. If I may reformulate your question, then you are essentially asking why this is the case.
In some generic situations, the answer can be simple. A very old argument, which was probably already known at the advent of statistical physics is the following (you might find it also in textbooks like "Statistical Physics Part 1" by Landau and Lifshitz). 
Take a body in thermodynamic equilibrium. Imagine this body now as constituted by $N\gg 1$ subsystems which are smaller, yet large systems on its own. If you consider these subsystems as "quasi-closed" you may write down an additive quantity $f_i$ which is well-defined already for each of the subsystems and calculate the average
$$\bar{f} = \sum\limits_{i=1}^N \bar{f}_i \sim \mathcal{O}(N).$$
The value of each $f_i$ may fluctuate around its mean value with a root mean square of $\left<(\Delta f_i)^2\right>$, with $\Delta f_i =\bar{f}_i-f_i$. If the subsystems are statistically independent, and if one requires the same independence for the r.m.s. one finds
$$\left<(\Delta f)^2\right> = \left< \left(\sum\limits_{i=1}^N \Delta f_i \right)^2\right> = \sum\limits_{i=1}^N  \left<(\Delta f_i)^2\right>\sim \mathcal{O}(N).$$
This implies, the relative fluctuations will decrease with the systems size as
$$\frac{\sqrt{\left<(\Delta f)^2\right>}}{\bar{f}} \sim  \frac{1}{\sqrt{N}}.$$
So far this is just the restatement of Timaeus answer. This argument can explain why we have such a thing as a well-defined energy. However, it fails to explain why there seems to be only a very limited set of emergent parameters that we need in order to describe all features of the system's thermodynamics. And someone may correct me if I am wrong, but this is still not very well understood in all its generality.
A few years ago, I came across the following publication: "Parameter Space Compression Underlies Emergent Theories and Predictive Models" (https://arxiv.org/pdf/1303.6738v1.pdf). The authors elaborate that this phenomenon of having a small set of some emerging "stiff" parameters which are sufficient to describe a complex system is a very general thing happening everywhere in nature and which actually enables us to do any science at all.
See also:
http://www.lassp.cornell.edu/sethna/Sloppy/WhyIsSciencePossible.html (authors webpage)
https://www.youtube.com/watch?v=cdaGIOFH8mQ (interview with the author)
A: That's just the point in a way, our probability based theories, both classical and quantum, are NOT FULLY capable of predicting future behaviour, only the probabilities of various outcomes. 
The power of statistical analysis comes from having enough data points to work from, so we get more and more confident about predictions as we have more data, but we can never be fully certain.
What are the attributes that hold for a system with many particles that allows us to study their whole behaviour as a system, with statistical physics?
The answer to this part of your question is basically, the same as above, it comes down to the more we know about the system, the better our predictions.
So at a  certain fixed temperature , I can roughly predict how a certain number of  particles are going to behave, at what frequency will their radiation be at its most intense say, but the more particles I have, the more accurate my prediction of this behaviour will be.  
The more information and the more particles we have available to study, for example  the motions of the particles, the initial values of the system, the forces that currently act (or may in the future act) on the system, all contribute to the accuracy of the predictions. 
Two examples to illustrate these points that  you might want to google are "chaos theory" which illustrates the importance of initial conditions on predictability , (or lack of it), and "maxwell velocity distribution".
Finally, I should say that in many cases, because we can't make totally accurate predictions, we often make simplifying assumptions, for example that the particles do not interact with each other, or behave in a simpler way than they actually do in real life.
