I see that I can modify the ring group descriptions under Phone System -> Ring Groups
But What I want to edit is the members of the ring group

You add phones to ring groups by configuring a PFK on the handset for the desired ring group. Personally I don't like this approach because there is not easy or central way to see what handsets are members of a given ring group. In small scenarios this is fine but for larger deployments I recommend using a system extension for this purpose.