Homebrew install permissions issue

I'd change the group permissions:

$ chgrp -R admin /usr/local/Cellar
$ chmod g+w /usr/local/Cellar

assuming your user account is in group admin.

You somehow have limited permissions to /usr/local/Cellar. Brew doesn't like to install with sudo which is why it refuses.

Check the permissions:

ls -ld /usr/local/Cellar

Open them up for writing:

sudo chmod a+w /usr/local/Cellar

Do not use sudo when working with brew (for security reasons).

You've to simple set-up your permissions.

So I would go even further and change the permissions to:

sudo chgrp -R admin /usr/local /Library/Caches/Homebrew
sudo chmod -R g+w /usr/local /Library/Caches/Homebrew

and then apply the specific group (either admin or staff) to user which should be allowed to use brew command. Check groups of your user via: id -Gn).

If there are further issues, run: brew doctor to see what's wrong.