The whole client is open source:

https://github.com/keybase/client

As far as I know, it asks for root because the Keybase filesystem runs on Fuse, and the menu bar app can be quit from its menu.

It's hard to balance making a simple, self-contained app with providing easy installation, a custom filesystem, automatic updates, and a command line tool… if you have ideas on how to improve things, I think they'd be open to issues/PRs. (I used to work on Keybase. I personally trust the devs, but, really, you can build from source if you want.)