Does psychology have a name for the "click" experienced in mathematics?
There are concepts like insight and the Aha moment which are similar to what you describe.

However, the aha moment is more aligned to flashes of insight to discrete problems. While this could be relevant to the experience you are describing of learning mathematics, perhaps you are also describing a more gradual process of learning. Mathematics can be quite complicated. With each iteration of practice, reflection and consolidation, an you typically learn more. At first you made need to go through the material just to get an organising framework. Then on  a second read, you can start to apply the organising framework to your interpretation. The meaning of the text and symbols begin to be stored in long term memory, so you can start to focus on the broader meaning rather than spending substantial cognitive resources on the components.

So in general, what you are describing sounds a lot like the process of cognitive skill acquisition. If you want to learn more you could read something like Anderson (1982). Basically, this is a movement from effortful and error prone performance to fluent and relatively error free performance. Thus, at a certain point, you reflect on your performance and you realise that you have acquired the skill. 

References


Anderson, J. R. (1982). Acquisition of cognitive skill. Psychological review, 89(4), 369-406. PDF