To put it simply, solving any mathematical problem involves analyzing numbers for patterns. Of course, complex problems involve several patterns that need to be recognized so that one can formulate the equations that predict the behavior of the problem being solved. To be able to recognize various patterns that crop up in mathematics, you need to expand upon your mathematical toolkit such as knowing the Binomial function or how to factor integers. For instance, when you learn something like the quadratic equation, you are adding a mathematical tool that you can use to analyze patterns in numbers.
In the end, It's really like putting a puzzle together except you know the big picture and are trying to find the equations that represent the puzzle pieces. The more propositions, lemmas, and theorems you know, the better you will be at extracting puzzle pieces that fit together to solve the big picture of the problem. For unsolved mathematical problems, you strip away all of the known patterns in the numbers until you have either solved for the equations that predicts the behavior of the entire system or are left with a pattern where you don't have the equations that predict the numbers. For the latter case, new mathematics can arise when you finally solve for the equations or algorithms that describe previously unknown patterns. If you cannot find the patterns that predict the behavior of the system, you can sometimes formulate conjectures on how these patterns might behave.
I would post an example based on my work on the Collatz conjecture, but I'm having problems getting latex to work in the new forum software. For some reason, the latex isn't being compiled into an image.