How do I keep macOS Catalina from automatically logging me out?

It's not logging you out, but locking the screen. To increase the delay before this happens:

1. Go to Apple menu > System Preferences > Security & Privacy.
2. On the General tab, adjust the dropdown for "Require password… after sleep or screen saver begins".

Of course, this can decrease security. If you're ever in a situation where you want to lock the computer immediately, click Apple menu > Lock Screen or press Control–Command–Q.

Go to System Preferences.

Security and Privacy.

Uncheck the box to require password after sleep or screen saver.

You can also go to Energy Saver and select "prevent sleep"

You may also have to go to Screen Saver preference and select "never" for the screen saver to start. But I think my very first suggestion should work without the others which would waste energy.