Hi Raphael,
"Dry up," means to take away moisture or to lose moisture, usually becoming withered or shriveled. For example, "The fallen leaves dried up in the sun."
It can also be used metaphorically. For example, to say that a person has "dried up" could mean that they are not very vibrant or very energetic anymore.
I don't have enough context to know what your example sentence means exactly, but I'm guessing it might be more of a metaphorical meaning.