Keybase is a website, but it's also an open source command line program. Let's walk through a terminal example, which illustrates what Keybase does. All of this can be embedded into other software, written by anyone - keybase.io
> keybase id maria pgp: C4B3 15B4 7154 5281 5100 1C58 C2A5 977B 0022 github: maria_leah ✓ https://gist.github.com/23423 reddit: maria_leah ✓ https://reddit.com/r/Keyba... twitter: maria_h20 ✓ https://t.co/maria_h20/523554 web: mariah20.com ✓ https://mariah20.com/X904F... ฿ bitcoin 1MPt9BuAVM6YphzyBCNUXkh5dprThwSvbD
The keybase command above looks up your pal, "maria", whom you know on github and twitter. In her case, you get back her usernames, which you recognize.
Satisfied, you may do spy-like things like encrypt a message to maria and paste it anywhere. GPG handles the crypto, using the public key verified above.
Keybase is an open directory -- no API key needed -- so you can request maria's key, get her proofs, and verify her identity in any software. The goal of Keybase is to let any security software be powered by usernames instead of offline key exchanges.
Again, you don't need to refer to maria by her Keybase username: you can put her twitter username directly into any keybase command, and the keybase client will confirm the public key is owned by that twitter user. The same goes for reddit, github, and other services.