The membership approval tab settings is used in conjunction with the builtin workflows "Approval by Primary Owner" and "Approval by Secondary Owner".
Three things need to be in place for this to require approval when a group is modified.
1. One or both of the above workflows are enabled
2. The accompanying option under "Membership Approval" tab for a group is checked (depending on which builtin workflow was enabled. For example if "Approval by Secondary Owner" workflow is enabled then the option "Approval by a secondary owner of the group" must by checked off for that group. This would be the same if the primary group owner option was used.
3. Under the "Manged By" tab, a user is listed in the "Name" field which would be considered the primary group owner or listed in the "Secondary owners" field.