On Windows:
Open a command prompt, and type:
adb shell
Then,
adb logcat | grep godot
On Mac/Linux:
Open a terminal, and type:
adb logcat | grep godot
This will show your console in realtime. To exit, just press CTRL + C.
I've seen some people say to use the appname, like your.app.name.
If you use just your appname, you will get info about your app, but you won't get any print statements. Because print statements are only prefixed with "godot" and not your app name for some reason.
What I do is try to include "godot" in my appname, so that I get the best of both worlds.
The appname is in Export > Options > Package > Unique Name
EDIT:
You can debug Android Godot apps inside the Godot Editor just like normal Godot games by checking:
Debug > Deploy with Remote Debug
Next time you run the project on your Android device, print statements and other info will show up in the Godot Editor.