"Be able to" means "can". But you can't use "can" with "might". So instead you say "might be able to".

You also use "be able to" when talking about things that you successfully did in the past:

I was able to contact Jesse, and he said he would send them to us right away.

Were you able to find out how much it costs?

And you use "be able to" with other modals like "should", "will", "may", etc."

He should be able to help you.

I won't be able to finish it until some time next week.