First, I really loved the film. The only question I have is for the keys with the note saying, "he belongs here with you." The part where the young, Hollywood bound runaway girl comes in to bake the cake. I thought maybe it was the blown up photo of Claire and her son, but then why the keys? Anyone?
shareHer Ex husband left the keys to the house on the counter. He came in and brought the picture of Claire and her/their son at the house along with the note. He knew the picture was what Claire needed. He hung it for her and then left the keys.
It was his way of saying "he belongs with you"..meaning their son would always be a part of them/their life together but he was saying he had gotten his clothes/things and was saying goodbye. He knew there was no hope of them getting back together.
In the script she didn't want to be around when he came to collect his things.