Just sign into Xanga, and visit a blogring that you've already joined.

See on the upper left side of the page, where it says:

Members:
» Invite Members
» Approve Members
» Manage Members
» Your Membership

Just click on "Invite Members", and a page will appear that lets you invite people to join your blogring.  To invite new members, just fill out the email fields on that page and click "Send »".