Teabing starts to monologue (because of course he does) while explaining why exactly he's holding them at gunpoint.
He'd vowed not to hurt anyone who hadn't betrayed the Sangreal. Wait, huh?
Oh. He's mad because he'd learned the Priory had decided never to reveal the documents to the world. According to him, Saunière'd eschewed his sacred duty by keeping the secret a secret after the End of Days.
Sophie is beyond angry to learn that Teabing's behind the death of her grandfather.
He says (and we paraphrase), "Oh, you're mad now? What if I told you the death of your family was a plot by the Church? Boom. Destroyed."
Teabing insists Sophie is only alive because she was handy leverage the Church could use against Saunière.
He further insists that now the three of them owe it to the Priory to reveal the Grail to the world, together. (Classic bad guy move, wanting the heroes to join the dark side.)
Teabing offers the cryptex to Langdon as a token of trust.
Well, it's official. Teabing is off his rocker.
Teabing thinks about everything he's done as the Teacher: betrayed Silas, killed Rémy (look, it was his fault for showing his face at the Temple Church). But hey: every Grail quest requires sacrifice.
He decides he doesn't need to tell Langdon and Sophie about how he's implicated Opus Dei in his plot. He can save that part for later.
He wants Langdon and Sophie to say an oath with him, pledging to find the grail and reveal the truth.
Sophie wants to say a four-letter oath at him, that's for sure.