Duolingo app's solution for this one was "The girls are bringing the book." How does one use vesz to mean "bring"?
It's visz here, not vesz.
Vesz has a pretty clear meaning of "to buy, take, receive". The object changes the owner.
Visz, on the other hand, looks like a bit more of a dummy verb. To take, to bring, to carry, these are the most common translations. It describes the act of transporting an object, and which of those translations applies depends on at which end of the way you're standing.
Because there is a definite object (a könyvet, THE book), you need the definite conjugation.
A lányok visznek könyveket - "The girls are taking books." The object is indefinite, and you use the indefinite -nek ending.
A lányok viszik a könyvet - "The girls are taking the book." The object is definite, and you need to use the definite -ik ending.
You can review the endings for the definite conjugation here (there are different versions for front and back vowel words, as usual): https://www.duolingo.com/skill/hu/Definite-conjugation
You can review when you need definite or indefinite here: https://www.duolingo.com/comment/17038425