Math is proof-based because they need to know that each step of reasoning is absolutely correct. For example in calculus they tell you what a limit is. But given some function or sequence, how do you know that the limit even exists? So in a course called Real Analysis, which you take after the first two years of calculus, they go back to square one. They give a logically rigorous construction of the real numbers, and they prove the least upper bound
property: any nonempty set of real numbers that is bounded above, has a least upper bound.
With that principle in hand, you can be certain that functions and sequences that "should" converge to a limit, actually do. Then they can make rigorous definitions of continuity and differentiability, give a rigorous definition of the integral, etc. So it becomes totally proof-based.
But you don't need to worry about that right now. However it's true that once you get past the first two years of calculus, the nature of math classes changes substantially. It's all definition/theorem/proof. But the concepts you study are very interesting in themselves, so don't let the idea of proof put you off.