a) They’re often playing material they’ve written themselves, and they remember how it goes 
b) Memorizing music, especially something you’ve written yourself, is no more difficult than memorizing the lyrics.
c) There is no sheet music. They musicians sat down together with their instruments in their hands and made up the song as they went along by playing what they heard in their heads, with a certain amount of give and take between the different musicians. Nothing was written down to begin with, or if there was it wasn’t sheet music - probably just a map of how the sections of the song were put together.
d) A lot of them can’t read music anyway.
That being said, my above points generally refer to actual bands. If you’re talking about a solo artist who lays down vocals on top of an accompaniment performed by studio musicians, then those studio musicians usually have some form of written music in front of them. Depending on the skill of the songwriter/composer, this written music could be as complete as a fully notated score (‘Weird Al’ Yankovic does this) which is ideally followed exactly, or as simple as a lyric sheet with the chord progression written in, leaving the studio musicians free to experiment until the desired results are achieved.
When the solo artist goes on the road he or she is backed by top notch professionals, and there is plenty of rehearsal before the tour starts to make sure everybody knows their parts. In that case, one of the backup musicians is designated the “musical director” (most often the keyboardist, or surprisingly often, the bass player, but it could be any one of them) and this person is responsible for, among other things, keeping things moving and properly timed and organized. The musical director usually has some form of written score with him on stage. These musicians are there to back up the solo artist, not so much to “perform” themselves, and most are quite disciplined (they have to be - they need to be ready, when Little Miss Pop Diva stumbles out on stage drunk, to make the show happen and make her look good
)
If an act requires an orchestra or similar accompaniment, the orchestra is indeed provided with sheet music. However, this is mainly because the artist doesn’t travel with an orchestra in tow — that would be prohibitively expensive! Rather, since most major cities have an orchestra in one form or another, the promoter books the local orchestra to accompany the artist when they come to that city.