My problem with many proofs is that notations are a bit complicated for me. Most of the time I don't even understand what a proof says or what a question wants me to do.
Prove that $S\subseteq\mathbb R$ is compact iff every infinite subset of $S$ has an accumulation point that lies in $S$.
I looked up what compactness is, but unfortunately didn't understand a single thing.
Your help and examples are much appreciated.
It sounds like you need to take a step back to learn and familiarise yourself with the basics. I would recommend searching for good online notes from some calculus/analysis courses or finding a well-reviewed book giving a broad overview of the area you're interested in. That way you can build a good foundation instead of learning bits and pieces on an ad hoc basis without seeing the big picture or motivation. This is not a criticism of your abilities; in fact, you have the most important asset, which is a desire to learn. Patience is key. Good luck with your studies!