Disclaimer: I am a graduate student in EECS, and not math. In particular, I am not a professional mathematician.
At some level, none of us is smart enough to understand higher mathematics (in its entirety).
Some illustrations (for which references are easy to obtain via a web search):
1. John von Neumann, in his time, said that he understood 28% of mathematics (no idea where he got that specific number from). Given the 20th century explosion of mathematics, largely driven by the professionalization of mathematics and the drive from the nation state, I can't think of anyone today who can truthfully answer > 5%.
2. One of the favorite, semi-secret pastimes of mathematicians is ranking other mathematicians, e.g is so and so a "first rate, second tier mathematician" or a "second rate, first tier mathematician", etc. The funny thing is that this ranking does not end: Fields medallists fall below the "inner circle" of Fields medallists (who e.g won the Abel prize as well), but even they look up to people universally recognized as singular, but who are no longer alive - popular favorites being Euler and Gauss.
A far more modest task is understanding some aspect of mathematics that one finds immensely fascinating. Fascination and enthusiasm are by far the most critical pre-requisites; without them one can't go far. Often what makes a field of mathematics difficult (at least in popular perception), say algebraic geometry for a stereotype, is the lack of familiarity with the language of it. Language takes time to seep into our brains. Some people can do this far faster than others, and hence appear as "geniuses" when they absorb things without much observable effort.
But that does not make it impossible for others; it usually just means more effort over a longer duration.
There is also a pattern to how mathematics operates. Enough exposure to a variety of topics will demonstrate that many famous, deep theorems, are at their core built out of (in retrospect!) simple, natural ideas - ideas so natural that one can come up with everyday analogs of them. In fact, in a highly simplified sense, most proofs at their heart involve what many would call "high school" manipulations (e.g a lot of arguments in real analysis involve what many call a "3 \epsilon" trick). Yes, there are subtleties involved, and a lot of effort sometimes needs to expended to complete arguments, but the same is true of various fields, including programming/software engineering (just look at the evolution of Unix from the original < 10k lines to today's *-nix).
If one wants something concrete that can be appreciated by many on this forum, consider:
1. http://www.math.ucla.edu/~pak/lectures/Math-Videos/comb-vide...
- Igor Pak's excellent archive of combinatorics videos; many of which are for a pretty general audience.
2. Federico Ardila's courses, see above for links; also see
https://www.quantamagazine.org/mathematician-federico-ardila...
If one checks the course websites of his offerings, there are tiny blurbs with some info about the students of his courses. Many come from quite non-traditional backgrounds, and a non-negligible fraction of them end up doing interesting research.
3. "The Princeton Companion to Mathematics" and/or its sister "The Princeton Companion to Applied Mathematics" - these books do a fantastic job of giving a bird's eye view of the essential unity of mathematics, and can be excellent starting points for diving into a topic suited to one's interests.
Lastly, mathematicians themselves do not always understand what they are doing/have done - otherwise generalizations/refinements that take many years to come up with would have happened much faster.
This is especially the case when they (in retrospect!) are breaking seriously new ground, and are thus probing far into the dark.
Gauss's earliest proof of quadratic reciprocity used induction on the primes; a technique, although useful, is far from most "modern" treatments; and is arguably not the best way to "understand" it.
In von Neumann's words:
"Young man, in mathematics you don't understand things. You just get used to them."