How do child care providers benefit from the adopt a grandparent program?

Everyone wins. The program matches individuals with elderly, sometimes nursing home residents for the mutual benefit of both the adoptee and adopter. Both get the joy that comes from a meaningful relationship with someone from another generation. A beautiful Swedish film years ago illustrated this "Kannst du pfeifen, Johannes" in which a child adopted a man in a nursing home telling him he was his grandson.
Many ways. Certainly the most benefit is reaped by the child. However, caretakers can get benefit from the program in allowing these "grandparents" to be role models in child-rearing, showing the importance of nurturing as well as teachers.