"Pick up" has the meaning in this case of beginning to do something that others are doing, particularly if it is stopping. A more common usage along these lines is "Pick up where I left off", which is something you might say when beginning to read a book in the middle where you had stopped reading the other day.
Macmillan dictionary includes this in its definition #11 of "pick up":
to start something again, from the point where you stopped
This is basically the same usage; it refers to the concept of beginning to do something, but not starting at the beginning. In the case of "pick up the chant", it is not you who earlier were doing the thing, but others.
I would suggest a subtle difference from some of the other answers here in my definition; it is fairly commonly used when the chant is dying down, more so than when it is gaining momentum. I would say "join in the chant" if it were still gaining, but "pick up the chant" or "take up the chant" if it were starting to die down.
'Pick up' is a very generic phrase in English, used for a very large number of different ideas all centered around (re)starting or gaining something or someone. "Pick up" a girl means to ask her out (on a date); "Pick up" a language means to learn it, but in a more informal way than taking classes; "Pick up" a radio station means to acquire a signal. The definition I linked above has 15 different usages, plus more sub-usages.