http://www.usingenglish.com/reference/idioms/with+flying+colours+(colors).html

Both answer above are correct. It's mean to pass exam brilliantly

It means, "extremely successful". The colors refers to the flags that might be waving in celebration of a victory. Common phrases are, "I passed with flying colors." or "I came off with flying colors".

"With flying colours" means with great success, to do something really well. I would say the most common usage relates to exams or tests. If someone does really well on a test, you can say they "passed with flying colours". I'm not sure of the origin, but my guess is it relates to parades or ceremonial processions which often would use multicolored flags or banners "flying" in the air.