"Predicting" something means saying what's going to happen, before it happens. For example, meteorologists try to predict the weather, and stock brokers try to predict the prices of stocks.
"I predict that ___" is a way to formally announce what you think is going to happen in the future. For example:
I predict that 2012 is going to be our best year ever!
I predict that he won't even show up to the first meeting.
When you start a prediction with this phrase, you seem confident.