You might start by studying basic abstract algebra, particularly with finite fields; basic number theory; and basic probability theory. You don't need to go terrifically deep into them—and beware elementary number theory, whose name is a cruel joke played by mathematicians on the unwary for the hardest way to do number theory.
You may also find it helpful to prime your mind with some of the ideas before you bang your head against them too hard, then see a bit of how they are used in cryptography, and then go back to them to understand how they actually work now that you have motivating applications in mind.
Once you have some facility with those basics and with mathematical reasoning, you may find Mihir Bellare and Phil Rogaway's class lecture notes on an introduction to modern cryptography helpful to work through some of the basic notions and proof techniques of modern cryptography in step-by-step pedagogical detail: interactive adversarial games, pseudorandom functions and permutations, security reductions, etc.
And, of course, you can also ask specific questions on crypto.se!