Hello aspirant, thank you for approaching us. Below given are few tips that might help you

1) Solve lots of problems - practice, practice, practice. Others can solve problems easily because they have seen similar problems before - much of problem solving is building off of experience.

2) If the problems seem insurmountable - you may be lacking the knowledge or experience needed to solve them. In that case take a step back to a simpler set of problems or the areas of math that are pre-requisites to what you are working on. [For example if Calculus problems are consistently too difficult - strengthen your algebra problem solving ability. If a proof about differentiable manifolds is killing you - check that your knowledge of Topology or of calculus on R^n is strong enough].

3) Solve simpler instances of the problem. Reduce the number of variables, dimensions. Make assumptions that make the problem simpler to solve - see if you can solve the simplified version. This may give you insight into the problem in its full complexity.
[If a proof is asking about properties of continuous functions - start with just looking at polynomials.]

Good luck!