In your example of a star coming towards a stationary body and falling, the energy comes from the gravitational field.
Gravitational potential energy is a function of space, its like a number each position has. So, before the star was there the gravitational potential energy had a certain value, I suppose it would be zero since there is no gravitational field strength (of course fields extend infinitely but it would at least be super tiny). I suppose its important to mention that nearby, say in a 1m radius, the potential is also zero. Agreed?
Gravitational force is the (negative of the) derivative of gravitational potential energy with respect to space. So since all around the stationary object there is no change in the potential, its derivative and therefore its force is also zero.
What would happen if a star approached? At some point the star would be close enough to a location near the object that that location would have a non-negligible value for gravitational potential energy. This would happen because the gravitational field strength would become large enough; in general it would get more non-zero the closer the star came.
So at the object you might have a potential of zero, but at the nearby location you have a potential of say -0.0005. Now all of a sudden you have a non-zero derivative, you have a force. The force is proportional to this difference; again no difference means the derivative is zero.
Just g by itself is linearly proportional to the source of the field's mass. So let's pretend these values were for a sun-like star, and the object was earth-like. Now we also pretend all of a sudden we wipe out the existence of the star with unnatural, god-like powers. What happens to the gravitational potential energy at our location? Nothing instantaneously, since no information can travel faster than light the changed situation (namely that there is no longer a gravitational field strength, no more star's mass) can't be known to us yet. This change has to propagate through the field, and does so even when the source is annihilated, and so the gravitational field has its own existence independent of the source.
Similar things happen with other forces. Really, energy as a concept is just a simplified way of dealing with forces and fields. Energy is a scalar, fields/forces are vectors. Scalars are typically easier to work with than vectors (compare arithmetic to vector calculus). Physically speaking, there is no physical mechanism in energy that "causes forces to arise" (at least that I know of). But for descriptive purposes it works out that the negative derivative of potential is equal to force.
You might check out this simulator. It is for charge and electric field, but can give you a familiarity with fields:
http://phet.colorado.edu/en/simulation/radiating-charge
But I should mention, gravitational waves have never been detected.