How to install Python on macOS Catalina using homebrew?

chown the usr/local directory:

sudo chown -R $(whoami) /usr/local

Then try to install python3 again and it should work:

brew install python3

OR,

You can try this also if the above solution didn't help

sudo chown -R $(whoami) $(brew --prefix)/*

Reference: https://gist.github.com/irazasyed/7732946