As explained on our getting started page, there are several important elements that should be included in your site:
https://wpgeodirectory.com/docs-v2/geodirectory/getting-started/#sidebar-widgets
Specifically, GD > Author Actions is the element that shows the edit/delete/add-franchise links.
It should be added to the archive item template and also somewhere on the GD details template, most commonly in the sidebar as a widget.
The default shortcode has an argument that the links should only be shown on the author page. This is a fine setting if you are using UserWP. If you place a list of listings for a user in another place, then that should be disabled.
You will also need to complete the setup of UsersWP integration to show listings on the profile, as described here:
https://wpgeodirectory.com/docs-v2/integrations/userswp/
Once that is done, though it is not mentioned there, you can visit your menu and use the UsersWP links to add links to profile and account.
If you have done those things, then please share some details about where you are seeing the issue.