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.