Why was loki adopted?

In the movie, it was because he was left alone in a temple to die and Odin couldn't leave a defenseless baby to die. The other Jotuns left him because he was small (a possible theory.

In Mythology, he was a small Jotun, so he ran away to Asgard where he and Odin became blood-brothers (they cut their wrists and mixed their blood, claiming to be brothers from that day forward). He later finds out he and Hiemdrall were swapped, so he is Odinson and Hiemdrall is a Jotun.

Hope this helps!