It means a pale colour that is closer to white (either closer to white than it was before, or closer to white than some other colour that is used as a comparison*). * In fact, in the song it is the former, since the couplet is "her face, at first just ghostly / turned a whiter shade of pale".
New words, one handy idiom, and a 2-minute quiz — delivered to your inbox to keep your streak alive.