In this case, it’s likely you’re a standard user and don’t have permissions to write to /Applications (which is now our default). If you never changed your default installation directory for apps and were using Homebrew Cask before we changed our behaviour to moving apps instead of linking, you should have some symlinks in your ~/Applications directory, which you do have permission to write to.
You can use --appdir=~/Applications when installing to bring back the old behaviour. For a permanent change, follow the Options section in USAGE.md. Specifically, you’ll want to set export HOMEBREW_CASK_OPTS="--appdir=~/Applications" in your shell’s startup file.
If --appdir doesn’t fix the issue or you do have write permissions to /Applications, the problem may lie in the app bundle itself.