I asked the same question and one of the participants explained that when followed by a post position, the noun gets the oblique case. In this sentence we have "to my home" and though "to" is implied, it still changes the case of the noun घर into oblique. Hence, मेरे घर.

I think अपने would refer to the friend's house rather than to your house. I think it would have to be अपने because there's an implied को after घर. But you need a second opinion because I'm just learning here.