What is rerolling?

I'm confused, people start over and over for what?

To get characters like Ace, Shanks, and Nico Robin because, they are only available through rare recruit and they dont want to spend money to get them.

So they would restart and redo the storyline over and over until they get good rare summons?

Pretty much. You run through tutorial stage, get you rainbow gem roll, if it is not who you want. Reinstall and start over again. I've seen reports of people doing this 100+ times probably taking days since this is the only time you have any control over the outcome of your roll.