It is another meaning of 한테 and 에게. It indicates possession if used with 있다, 많다, 없다, etc.
Think of it like "have something WITH/ON me".
- 저한테/저에게/제게 그 책이 있으니까 내일 갖다 줄게요. = As I have that book with me, ...
- 제가 그 책이 있으니까 내일 갖다 줄게요. Same meaning. Both 한테/에게 and 가 are natural.
- 저는 그 책이 있으니까 내일 갖다 줄게요. Unnatural. The point of the sentence is bringing the book to the other person, so there is no need to draw attention to 저 by using 는.