You actually say "I'm sure (something is true)" when you are not completely sure. You use it when you're guessing about something. The level of sureness is similar to "probably". But saying:

They probably won't mind.

sounds a little more careful, while:

I'm sure they won't mind.

sounds more confident.

When you use this phrase in writing, you should use "that":

I'm sure that it's perfectly safe.