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.