Ok so you can pronounce it 2 different ways. You can pronounce it as "ruh-ig", which is the most correct or "ruh-ich" which is slightly slang, but it's used all over Germany. Even Hoch Deutsch speakers will pronounce it with the "ich" sound, however, both are correct.
"leise" is the opposite of "laut" (loud, noisy). It is not completely silent, that would be "still". "ruhig" comes from "Ruhe" (calm, rest), so it means calm, relaxed. When you are calm, you don't make much noise, so ruhig implies leise.