> If I don't want to give keybase my private key, which I obviously don't, how can it sign my tweets?

You don't sign all of your tweets. Just one which proves that the person who owns the keybase account also owns the twitter account in question. Through transitivity, you can then prove that you are the same person who owns a particular facebook/github/HN account if you also sign a post on those services.

> Presumably if I can access my account, t's me.

That proves the person logged into the account is authorised to log in, not that the owner of the account is a particular person or the same person that owns another account on another service.