I think my first leap came when I realized mathematics is not about the formulaic plug-and-chug exercise type of mathematics commonly learned in grade school (particularly in the USA). E.g. x^2 + 1 = 5, x = -2,2. Great, now repeat this 100 times.
So then what is it about? Well it is about going deeper, asking why does this work. Instead of focusing on specific statements like solving the above problem it is about conjecturing what is the technique to solve all such problems (e.g., quadratic formula). And then proving that this technique works for all given numbers of a certain type.
After that it is about thinking even more deeply about the problem and saying, well does this work only for real coefficients? what about complex? what if we restrict ourselves only to remainders mod 5 (e.g. the numbers 0-4 mod 5) and so on. Why only polynomials with powers of 1,2. What about 3,4, and above? What about polynomial equations in more than one variable?
More broadly it is about appreciating the challenges experienced before the theory to solve some problem was developed. Then understanding that some new innovation was required. Generally, this takes the form of defining some new type of object, showing that the problem we are solving is but one instance of that object, and then saying, well based on these definitions and what we already know, here is what we can say about these objects. This is the process behind building a theory.
I think when you can start appreciating some of the ideas that the mathematics is saying, and the limited understanding of the world before that theory was developed, and then following a similar process to develop a mathematics of your own, then you have matured as a mathematician.
This is but one humble attempt to answer the question briefly.