As an example: you'd reasonably expect a student with mathematical maturity to know what a set, the Cartesian product of two sets, and a function are. That's high school level math.
From there you can define a ring and a module over a ring, in like 5 mins.
So at that point the Prof, strictly speaking, told you everything you need to know to start studying modules.
Of course, many students will be like whaaaaaa?! When a ring is defined. But here's the point: Mathematical maturity does not mean that you already know what a ring is, but when the Prof defines a ring, you are following along.
Anyway, my point is that there is nothing specific that you need to know.
Some people are better with math than others. Undergrad math majors thought the major was pretty easy. Many of us in engineering majors--who weren't that bad at math (it was a good school)--could no more have graduated with a degree in math (or physics) than have flapped our arms and flown.
Any good definition of mathematical maturity would likely be so technical that you need mathematical maturity to understand it.
Most colleges have classes designed to foster mathatical maturity, but that is done by how the class is taught, not what is taught in it.