Why did everything in space cool out? Through my research, I learned that;

According to thermophysics, heat always moves from an area of high
  heat to an area of low heat. Space has no heat at all. It is extremely
  cold

However, if I am not mistaken (I am not a physics guy), based on the Inflation Theory, the spaced expanded extremely fast. This is how physicists can explain the nearly exact same heat all over the space. What I am asking is:


*

*How inflationary theorists explain the same heat all over the space? Is it because that the total heat was scattered everywhere evenly and then cooled the same amount? If so, how come all space cooled down the same amount of heat?

*After the inflation and expansion, why did the hot plasma cool out? What is the reason?

 A: Ever heard of the cosmic microwave background?
The CMB is a relic from when the universe became "opaque" - when, as Wikipedia says,

protons and electrons combined to form neutral atoms. These atoms could no longer absorb the thermal radiation, and so the universe became transparent instead of being an opaque fog. 

So photons decoupled and the CMB was formed.
Curiously enough, the theory of inflation was posited to explain why the CMB was the same temperature throughout space (it also explains why, at the largest scales, the universe is homogenous - it appears the same).

After the inflation and expansion, why did the hot plasma cooled out? What is the reason?

I assume you're talking about why the CMB cooled down. The answer is that the universe is expanding, and, therefore, the photons of the CMB are redshifted - just like how photons emitted from a source moving away are redshifted. As the photons are redshifted, they cool, and so the temperature of the CMB drops. If you're interested, this formula may be interesting:
$$T_f=2.725(1+z)$$
where $T_f$ is the temperature of the CMB and $z$ is the redshift. You can use this to calculate the temperature of the photons at any point in time - or, alternatively, if you know the temperature of the CMB, you can calculate the redshift at any point in time.
