You actually say "I'm sure (something is true)" when you are not completely sure about something. You use it to announce a guess:

I don't think we need to book a hotel ahead of time, though. I'm sure we can find somewhere to stay.

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.