Say you're given a paintbrush and a really tall ladder and asked to go up to the ceiling of the Sistine Chapel to make some "improvements."
That's pretty much what composer Thomas Newman felt he was being asked to do when he was charged with scoring The Shawshank Redemption. He thought it was a masterful work, and was scared to death he was going to screw it up.
For that reason, the soundtrack is mainly unobtrusive. There aren't many moments where you think to yourself, "Ooh! Pretty music!" Instead, it's very much in the background, and is just there to help set various moods (usually depressing ones, although at times soaring and uplifting), and not to interfere more than it has to.
However, one reason the music is so important in Shawshank is that there are roughly a bazillion montages. A montage needs music. Without music, it's pretty much just a silent flipbook. No one wants to watch/read a flipbook. Let alone a silent one.
While the incredible writing certainly helps to draw out of the audience the intended emotions and reactions, the film's score does nothing but add to it, so you can rest easy, Mr. Newman. Not that he really should have been worried. This is the same guy who contributed his talents to such amazing films as Wall-E, American Beauty, and Saving Mr. Banks, so he knows a thing or two about not "screwing up." Plus, he gets a helping hand from Mozart's opera.