Nice to see Lightning slowly getting Taproot-ier. From what I understand this change makes it so that the on-chain funds can be claimed by co-signing a transaction between the sender and receiver. Since Taproot makes it possible for a 2-of-2 multisig to look and be as cheap as a singlesig address, this leads to more privacy.

It is still trustless, so as long as you have the preimage to the HTLC you can still claim the on-chain funds. But again Taproot makes this even more private, you would only need this alternate spending condition (or the condition for when the swap times-out) when you need it, if for example your peer doesn't sign the transaction with you instead of having to provide the entire script when spending (cheaper and more private yet again!).

It's amazing how at once these new innovations make everything cheaper. Less block space, less information leakage, less processing. What a time to be alive.

Suggest where should I go to see to understand taproot?

This is always good news. I think one of the main reasons for Taproot was Lightning. I don't know more technical details but I am overwhelmed with all advancements everywhere. It is being very productive cryptowinter.