Apparently the application code for the IOS app is open source and is located here.
It seems they use the appropriately named Signal Protocol.
According to wikipedia,
The protocol combines the Double Ratchet Algorithm, prekeys, and a triple Diffie–Hellman (3-DH) handshake, and uses Curve25519, AES-256 and HMAC-SHA256
On the iPhone they are probably using one of these repositories.
WhisperSystems/SignalProtocolKit
WhisperSystems/libsignal-protocol-c
As far as explaining how the underlying cryptography works, I would not be best suited to do that.
