"전 얘로 할게요." is abbreviated sentence.
1. 전 -> 저는
2. 얘로 -> 이 아이로 (this child)
Although birds is an animal, childs may think the animal as valuable as if it is human.
So, I assume that one child tell one of the birds as this baby
3. 할게요 -> (선택)할게요.
So, "저는 이 아이로 선택할게요." is abbreviated to "전 얘로 할게요."
, and the meaning is that "I would like to choose this baby."
Considering the situation, one child say, " I would like to choose this baby(bird)."
And then, the other child will choose one of other birds."