Due to the non-discoverable nature of a Private Group, Membership Applications are not available on a non-signed in public web page. As such the main way to add members to a Private Group is through Invite links.
You can create and copy Invite links through the Manage Group admin area. These links will expire 7 days after creation.
When using an Invite link for a Private group, the Join Group form will lack any kind of group website chrome.
Using this form people can create a Member account and join the group.