In order for an email template to be selected for use in a campaign, it must be Published. You can see the status of a Published email template by the green checkmark in the upper right corner when viewing Settings > Email Templates:
When a template is still in a Draft state, it will have a blue ribbon displayed instead:
These templates cannot be viewed from within campaigns, and must be published first. If the folder containing the template only has draft templates, the folder itself will not display when editing a campaign also.
You can publish a campaign by hovering over the template, pressing edit, then in the bottom right corner press Save & Publish:
Once your template is published, it will have a green ribbon displayed and should be able to be viewed and selected during the campaign editing process.