。
It looks like next week's going to be busy.
Literal
Next-week [topic-は] busy-become-seems-to.
忙しくなる = 忙しい (busy) adverbial く + なる ('become') — 'become busy.' Then ~そうだ on the masu-stem (なり → なりそう) renders visual inference 'looks like.' Combined: 'looks like it will become busy.' A neat stack of two productive change-of-state and conjecture patterns in a single short line.