There are lots of ways to partition one's mathematical ability into stages to gauge your progress and readiness for something deeper. The stages are not strictly partitioned, so it is possible to kind of move back and forth between different stages, and you may be at different stages with different areas of study. Here is one such way that applies in this context.
In the first stage, one learns how to compute. There is no theory, no generalities. There are numbers, equations, matrixes, etc, and at this point, the student's sole job is to get good at moving them around. This is the portion of your mathematics upbringing from when you were a small child learning to count, up through whenever you started doing serious mathematics.
In the second stage, you learn what proof is and how to construct them. You learn how to prove standard theorems, and every class you take will prove every theorem, except possible for some very difficult ones that are outside the class. You are expected to learn how these proofs go and how they relate to one another, and maybe even are asked to reproduce them. Most people do this for pretty much all of undergraduate.
In the third stage, instead of learning proofs, you now learn proof techniques. By having a large swath of ideas at your disposal, you no longer need to see every detail of a proof to know how it works (unless it's really technical). When you see a theorem, you don't just rattle off the steps of the proof you have memorized. You think about what kinds of ingredients go into the proof, and you stitch them together. This is not to say that you will be able to reproduce every theorem you ever learned flawlessly - of course people forget things all the time - but it means that knowing what you know about the subject, you can rediscover and reassemble the proof on the fly, even if it means you trip from time to time.
I would say that in all probability from your question, you are in the second stage, and asking about what it means to go into the third stage. Do you have to know every single argument to every theorem to be a good mathematician? No, of course not. Learning every proof to every theorem would take years, and you would never get to advance. On the other hand, the only way to get from the second stage to the third stage is to learn as many proofs as you can, and to analyze them thoroughly. Find their crucial steps, the fundamental things that hold them together, and reflect on them. Then, advancing to the third stage is all about looking for ways to use those fundamental observations for other problems.
Hope this helps!