It might feel like the chain of citations goes ad infinitum, but of course this can't literally be the case. Indeed as long as mathematics is (at least primarily) an endeavor by and for humans, the entire chain of reasoning must be graspable by about the time a mathematician reaches PhD level. Much of it will not yet be accessible to an advanced college student such as the OP, and modern mathematics has gone in so many directions that no one mathematician can grasp more than a small sliver of the frontier $-$ which is why modern research mathematics requires specialization, and a mathematical community large enough to support such a large frontier. Inevitably errors occur, and a few of those propagate for some time before being caught. But the enterprise as a whole is self-correcting (as already explained in several ways in other answers).