In this case, it’s likely your user account has no admin rights so you don’t have permissions to write to /Applications (which is the default). You can use --appdir to choose where to install your applications.
If --appdir doesn’t fix the issue or you do have write permissions to /Applications, verify you’re the owner of the Caskroom directory by running ls -dl "$(brew --prefix)/Caskroom" and checking the third field. If you are not the owner, fix it with sudo chown -R "$(whoami)" "$(brew --prefix)/Caskroom". If you are, the problem may lie in the app bundle itself.
Some app bundles don’t have certain permissions that are necessary for us to move them to the appropriate location. You may check such permissions with ls -ls <path_to_app_bundle>. If you see something like dr-xr-xr-x at the start of the output, that may be the cause. To fix it, we change the app bundle’s permission to allow us to move it, and then set it back to what it was (in case the developer set those permissions deliberately). See litecoin for an example of such a cask.
Help us by submitting a fix. If you get stumped, open an issue explaining your steps so far and where you’re having trouble.