"Over and over" is similar to "again and again". They both mean "many times". You repeat "over" two times because to "do (something) over" means to do it again one time. To do it "over and over" means to do it many times.

"Again and again" and "over and over" are similar in meaning. But "over and over" sounds like something that's more work and harder to do than "again and again":

What wears you down is when the same mistakes get made over and over again.

I keep making the same mistake over and over.

A good speaker understands the power of a simple message that is repeated again and again.