For an introduction to adding fonts, see the Activate new fonts on your desktop tutorial.
Fonts are activated on your computer through Adobe Creative Cloud desktop. If you do not have the software installed, download it here.
Creative Cloud desktop must be running and you have to be signed in to Creative Cloud for the fonts to be activated. If the software isn't running or you have signed out of Creative Cloud, the fonts will be unavailable temporarily.
Make sure that you are signed in to Creative Cloud with the same Adobe ID that you use for your Creative Cloud subscription.
Your account information is at the top of Creative Cloud desktop; click your profile icon to reveal the name and email address of the account you're using.

To switch to a different account, click Sign Out. After you are signed out, you will see the option to sign in with another Adobe ID.
You can also click on the “Account” option in the Creative Cloud desktop profile menu to visit your Adobe Account page, and confirm that you are signed in with the Adobe ID that has your Creative Cloud subscription on it.

The Fonts menu is updated when the font is active. A red dot will appear next to any font with an activation issue. Hover over the dot for more information about the issue.

If a font fails to activate or there is a conflict with a font already installed on your computer, an error message is displayed with information on how to resolve the problem.
Pozor:
If the Fonts menu is a blank white window or shows a loading spinner, refer to Creative Cloud desktop app doesn't open | Progress wheel spins continually for further troubleshooting.
If the fonts are not active, try turning off the font option in Creative Cloud, wait a moment, and then turn it back on.
Open the menu from the gear icon at the top of Creative Cloud desktop. Choose Services, and then toggle Adobe Fonts to turn it off and back on.

Signing out of Creative Cloud and restarting the computer stops any font-related processes that may not be running correctly and starts new ones.
Windows 10 has an option to "block untrusted fonts" which, when turned on, also blocks Adobe Fonts from activating.
This article from Microsoft explains how to check whether it's on for your computer:
If block untrusted fonts is on in either the Group Policy or the Registry Editor, it needs to turned off or changed to the "audit" setting for font activation to work.
If your network proxy or firewall blocks access to any of the required network endpoints, Adobe Fonts will not work. Your computer needs to be able to reach these endpoints and download files from them.
Here is a list of the required domains; port 443 also needs to be open.
https://data.typekit.net:443
https://state.typekit.net:443
https://polka.typekit.com:443
https://api.typekit.com:443
If these steps don't solve the problem, contact Adobe Customer Care for further help.