"better" is an adjective or adverb that you use to compare things.
That is a better size.
He runs better than I do.
"better off" means "in a better situation". It is like a state.
Because you are sick, if you stay at home, you will be better off.
==> Because you are sick, if you stay at home, you will be "in a better situation".
"better off" is used a lot when talking about finances.
If you get a raise at your job, you will be better off.
==> If you get a raise, you will be making more money and your financial situation will be better than before.