라고 is almost like the English word "that." It's used for quotations.
I don't really know a good way to explain it because it can be used in quite a variety of situations, so I'll just give you a few examples.
1) 저는 (name) (이)라고 해요. I am called (name).
2) (thought) (이)라고 생각해요. I think that (thought).
3) (speech) (이)라고 했어요. (He) said that (speech).
The (이) is added if the preceding syllable ends in a consonant.