Musical theatre is a curious creature. It combines composed music, which is mathematically precise—the value of a quarter note is the same no matter how many times you play it—and Theatre, whose practicioners can be counted on to read each line slightly differently from night to night. Obviously, some sort of kludge would be necessary to combine the two, and so the vamp was born. Let's first look at how a vamp might look in a musician's score, and then talk about how it works.
/ \
/ Vamp (voice last x) \
/ \
------|-|----------------------------------------------------|-|------------------------
| | ___ | | /QQQ\
------|-|-------------{-------------------------/QQQ\--------|-|-----\QQQ/--------------
2 | | . / \QQQ/ . | | | ...
------|-|------------{--------------------------|------------|-|-----|------------------
4 | | . \ | . | | |
------|-|-------------}-------------------------|------------|-|-----|------------------
| | | | |
------|-|----------------------------------------------------|-|------------------------
\ 37 / 38
\ /
\ /
SARAH: When you ...
Okay. As a musician, your first warning sign that this is a vamp are the big, stylized brackets and the words "Vamp (voice last x)." This means that when you arrive at measure 37, rather than going on as usual, you keep repeating the measure (the quarter rest and quarter-note C).
This measure will repeat until the character named Sarah has finished her dialogue (or has reached the correct location on the stage, or finished opening the letter, or any other of the thousand things that actors have to do) and sings the word "When." ("Voice" in the context of vamping notation always means people singing. If the composer means people talking, it will say "dialogue.") The word "when" will come at the same time as the quarter-note C, because the word is printed directly underneath the note.
When this happens, rather than repeat back to the beginning of measure 37, proceed on to measure 38 and play the quarter-note D, which will correspond with Sarah singing the word "you."
Of course, it's rarely as simple as that. More often than not, only the conductor's score will have the actual dialogue written in underneath the score (as I have done), leaving the other musicians in the dark.
Also, keep in mind that traditionally, the musicians face away from the stage, meaning that only the conductor can see both the orchestra and the stage. This means that the poor conductor is responsible for cueing the musicians and the actors (and, when there are serious personnel shortages, playing the piano). so sometimes the cue you're looking for (a raise of the hand, or a large head movement if the conductor's hands are otherwise occupied) will get lost in the shuffle. Also, the vamp instructions won't always be as clear as the example; sometimes it will just say "vamp until ready" (Ready for what? Who knows?), leaving you even more at the mercy of the conductor's cueing. But this is what rehearsals are for, and with luck, the orchestra can get all of these questions cleared up long in advance of opening night.
One more thing you may run into: Due to whatever staging decisions the director/choreographer has made, an amount of time (to get ready, to get through dialogue, to make an entrance, &c) that the composer thought was reasonable turns out not to be. This forces the actors to rush in order to make their song cue in time, and is guaranteed to cause ill will all around.
If this is the case, the conductor, with the blessing of the director or choreographer, can write in a new vamp, creating more time for the actors to get ready. If this happens, the conductor will say something like, "Okay, so we're going to vamp on measures 45 through 49 to give him time to get all the way downstage. Watch me for the cue." You would then write this in your score (with the brackets before measure 45 and after 49, and the instructions), and then everyone should be happy.
This, of course, is why any sort of theatre more advanced than elementary school uses a pit orchestra, rather than pre-recorded music. Simply too many things can happen to get the performers and the tape out of sync, and having a real, live person in charge of the music allows more flexibility.
Source: my own experience as a first-time member of a pit orchestra, and my need to get up to speed very quickly on things like vamps.