"I'm terribly biased" means "I am heavily in favor of one side over any other, maybe to a fault." For example:
"I'm a huge fan of FC Barcelona, so I'm terribly biased if you were to ask me which football team is the best." In this example, the person is admitting that, if you wanted to know what he/she thought was the best football team, he/she would not be able to give you a neutral answer because he/she is already a huge fan of one.
"My entire family lives in Israel; therefore, I'm terribly biased when talking about the Arab-Israeli conflict." In this example, the person is admitting that he/she already favors one side (and gives you the reason why) when any discussion about a major world issue comes up.
In both of these examples, the people are admitting that they have a fault: they cannot look at these specific issues neutrally because they already favor one side.
I hope I've answered your question. Please let me know if I haven't.