Why are general and special relativity important to the field of astronomy?
General theory of relativity has more to do about Astronomy than special theory. It helped us explain the precision in the orbits of many planets we observe.
Unlike most people think, general relativity has nothing in General in a sense, neither do the special relativity that has something 'special'.
Like Newton's laws, General relativity makes its starting point as follows :
1. Speed of light is constant across all frames of reference
2. Effects of Acceleration due to gravity and Acceleration due to a force is indistinguishable (this is quite not clear and sketchy at best)
3. Laws of Physics are independent of the frames of reference.
Making these as the starting points, Einstein extrapolated what might be possible scenarios these might lead to if assumed to be true. In a little detail, since space is dilated due to relative change in speeds, and since acceleration causes continuous change in speeds, acceleration should cause continuous dilation in space. Also as acceleration might change, so does the dilation of space. So, space becomes an active player, not a passive stage on which motion is observed.
Result : Going by the second assumption of Einstein, we can say that since gravity changes with height causing change in acceleration uniformly and "continuously", gravity might cause any amount of space in its vicinity to continuously dilate or bend towards its inside.
Applications to Astronomy : Since space is no longer a passive player, we can assume pushing space to its extreme i.e. a complete and heavy bending on the space or a kind of collapse on itself - like a crushed paper being crushed more and more all the time. That extrapolation is what we call 'Black Hole', whose discovery is very recently established and Einstein proved right, which means theory might be right.
Most important of all, it explains the location for possible masses, which might not catch our attention, by explaining the observed change in its surrounding masses. Thus we discover new planets, explain galaxies, formation of new stars and Big Bang itself !