It's nice to see that someone has written a Catalina-compatible app to replace the old 32-bit apps APOD Grabber, APOD Viewer and AutoAPOD. Please note, however, that this app is not free. After a 10-day trial period, it is $10.00/year.
While that is a small sum, if you wish to save yourself $10/year, you can always go to https://apod.nasa.gov/apod/astropix.html and manually download the images yourself. On that page, below the actual current image, you will find links for "Archive" and "Calendar". Check those links out as well for previous images.
If you want to use the images as Desktop images, after manually downloading them from the above URL, just place them in /Users/your-account-name-here/Library/Pictures and then select the desired image in the "Desktop & Screen Saver" prefs pane in the "System Preferences" app. In the prefs pane, you will find your APOD images in the "Pictures" folder.
If the above instructions are too complicated for you -- it's really not -- just buy this app then. :)