There would first be 1 broadcasted on-chain transaction setting up the channel, with X of my bitcoin and Y of your bitcoin being locked up. This prevents double spend. Then, you and I pass n signed transactions between each other updating our balances Xt & Yt for each t, t+1, ...t+n. At any time, either one of us can broadcast the final transaction with the most up to date balance and unlock the coins.

So it requires one on-chain transaction per pair of parties instead of one per transaction? Clever and I suppose useful but seems a little limited in use, given that the funds are locked the whole time. Good for two parties that frequently have transactions going in both directions?

That's right. You can also form a chain of these channels, so Alice can pay Bob even though she doesn't have a channel with him, if they both connect to Carol using atomic transactions that update the 2 channels accordingly.

Is it that the not-on-the-Bitcoin-blokchain account gets updated on a separate "layer 2" Blockchain that's cheaper? If so, why would that second chain stay cheaper?

Lightning transactions are all final and irreversible via cryptography mechanisms.

Theres a timelock enforced by the blockchain.