To perform an atomic swap, both assets are locked up in a 2-of-2 multisig address (with a timeout that allows the original owner to reclaim it). At least one of the two networks has to support Schnorr signatures (preferably both). Then, a transaction is created that transfers the funds to the counterparty using an "adaptor signature" on one of the networks. The latter is a signature that is "damaged" with a scalar offset which the signer proves to be valid except for such an offset with a zero knowledge proof. The other party creates responds by creating a transaction on the other network, proving that its signatures is damaged by the same offset.
If either party now uses the knowledge of the damage to fix the signature, the other party can calculate the original damage and fix the other transaction.

Due to the Schnorr's compact multisignature the 2-of-2 multisig address with the timeout fallback looks like a standard 1-of-1 address, and the 2-of-2 transaction spending from it also looks indistinguishable from any other standard transaction.

In result, Schnorr allows to perform atomic swaps by creating two standard transactions on both networks, whereas previous atomic swap constructions made use of 2-of-2 multisig addresses and HTLC contracts which were much heavier and easily distinguishable.