It's pretty subtle. In general, you can just say "I think that" without changing the meaning of the sentence significantly.

"I do think" can precede a qualifying statement. "I do think she's attractive, but I don't want to date her."

It can also be used for emphasis, especially when you know that your opinion is unusual or socially unacceptable. For example: "Yes, I do think it's okay for parents to abandon their children if they get tired of them."  

Probably the most important usage, and the only case in which you really ought to say "I do think that", is when your opinion has been questioned.

For example: "you don't really think that, do you?" "yes, I really do think that."

Or,

Stealing is wrong, and thieves should be punished, but I do think that chopping off someone's hand is a bit harsh.