Gravity, falling bodies, and the equivalence principle Why is that bodies in a box accelerating uniformally in space that is sufficiently removed from gravitational fields fall identically to bodies in a box located in a homogeneous gravitational field (e.g., on Earth)
 A: We don't know. The equivalence principle is based on a fundamental assumption that interial mass is the same as gravitational mass. We have verified this assumption in numerous experiments (e.g., the classic feather vs hammer experiment done on the Moon by David Scott). 
On this assumption, we have also built a more complete theory of gravitation (GR) that explained/predicted thitherto unexplained/unknown phenomena (e.g., black holes, gravitational waves, gravitational lensing etc.). It all works out and seems correct but we still do not know why the two masses should be equal. For all we know, there is a deeper connection lurking in somewhere.
A: If you have ever taken an airplane flight, you will remember at the time of take off it accelerates and you feel how your body gets "stuck" to your seat. And if you try to lean your upper body towards the seat in front of you, it takes some effort and if you relax your muscles you "fall" back down to your seat again. Notice this is exactly the same effect as when you are standing up on the ground, and you use the muscles of your legs to jump up. The force of gravity pulls you back down.
Now imagine you were in empty space, sufficiently away from any gravitational field so as to feel weightless, and you are inside a spaceship that is about to take off, but unlike the airplane on the Earth, the spaceship accelerates vertically upwards instead of horizontally. Therefore, instead of feeling a force towards your seat as in the accelerating plane on the Earth, you feel a force downwards, ie towards the floor of the spaceship. And it actually turns out that the spaceship is accelerating at $9.8 m/s^2$, and it has no windows for you to look outside. Now imagine your memory of going into this spaceship was deleted, and immediately after you were told that you are standing on an spaceship which is sitting on the surface of the Earth, would you believe it?
A: In one case a force is inducing an acceleration on the mass, while in the other (gravity) an acceleration is inducing a force on the mass. 
In both cases the relationship between force and acceleration depends on the same interaction between the mass and the cosmological metric through which it cuts, and the relationship is therefore identical.   
A: This is called (as you mentioned) equivalence principle.
Actually, Inertia and gravity are same at fundamental level. There is only one mass, we just experience it as two equivalent ones - Inertial, and Gravitational. The distinction is not really necessary.
Inertia - A body due to its own mass (energy), creates a dip of space around its center of mass. That dip makes a force be required to make a change in state of the body. We call this property inertia.
Gravity - Same dip (curve) due to mass (energy) of the body manifests as gravity for other bodies. (And for itself as well, which gives the body round shape given sufficient mass)
So, their origin is same and that is curving of space.
Inertia is nothing but gravity of the body acting on itself against any change of state. This is what is happening in the accelerating lift example - the lift accelerates but the ball stops accelerating as soon as you drop it in the lift. That is why, it will have same impact as it would have had under same strength of gravitational acceleration.
My view is that gravity and inertia are same phenomena. They are two sides of same coin.
