I would consider there to be three main reasons for having an invite-only social media site:
1) Exclusivity fosters more excitement for users than if just anyone could join. It is classic marketing practice - creating scarcity in order to drive demand for it.
2) Invite-only sites promote valuable content. For example, Dribbble presents only design work from members who requested in and were accepted for their talent.
3) Stronger connections are forged. Users often may only sign up when they know someone who is already joined up, thus fostering stronger and more valuable relationships between members as they already know each other.