Course groups that are displayed on Join Course Page must have the following settings (Learners and groups>Group Settings):
1) Allow requests to join/leave this group? - Yes
2) Auto-accept requests? - Any value (Yes or No)
If Auto-accept requests set to No you must have valid outgoing email settings.
If group has setting Auto-accept requests? - Yes then user sees the Join link on the join course page. And after he clicks it he is added to the appropriate group immediately.
If group has setting Auto-accept requests? - No-> user sees Send Request link to the join course page. And after he clicks it his request is sent to mailbox taken from the group settings.