I think the best way to explain attachment to someone else, especially someone who is going into a very similar situation, is to say that it's normal to have difficulty at first. In creating a relationship with someone new—in my situation, it was my parents and siblings—it's very difficult to wrap your head around something that big and to understand, “Holy crap, I have a family now”.
Being able to just have time to process that is important. Part of the attachment, of course, is processing the idea of “Oh my gosh, someone wants me”. It's also about being able to build that relationship and realize that these people care about you. It's building that trust, being able to care back and being able to build that love and that relationship.
It goes with any kinship situation. It could be a family friend that you consider family. It's the same thing. If your parents were to bring in a friend and they said that this is their really close friend, you build a close relationship with that person and maybe you get close with them. It's the same idea, but instead of it being someone who is like family, they become family.