Can I rename files in a directory with Vim?

Yes, it can't be done like that.

That "file-browser-type interface" is provided by a built-in plugin called netrw. It is read-only so yeah, you can't modify it.

You are supposed to hit R to rename the file under the cursor or the marked files.

See :help netrw, :help netrw-browse-maps and more specifically :help netrw-R.

If you want to batch-rename files using Vim you should try qmv from the renameutils package or vidir from the moreutils package (thanks to Dmitry for the heads up).

Check out the renamer.vim - Use the power of vim to rename groups of files plugin (now purely maintained on GitHub). Like netrw, it presents the directory contents in a scratch buffer, and then lets you edit that buffer, and finally apply the edits to the underlying files.