How do mathematicians conduct research?

I am curious as to how mathematicians conduct research. I hope some of you can help me solve this little mystery.

To me, mathematics is a branch where you either get it or you don't. If you see the solution, then you've solved the problem, otherwise you will have to tackle it bit by bit. Exactly how this is done is elusive to me.

Unlike physicists, chemists, engineers or even sociologists, I can't see where a mathematician (other than statisticians) gather their data from. Also, unlike the other professions mentioned above, it is not apparent that mathematicians perform any experiments.

Additionally, a huge amount of work has already been laid down by other mathematicians, I wonder if there is a lot of "copy and pasting" as we see in software engineering (think of using other people's code)

So my question is, where do mathematicians get their research topics from and how do they go about conducting research? What is considered acceptable progress in mathematics?