Atomic swaps realize cooperation between several two blockchains in the sense that a transaction on one chain can be executed if and only if another transaction is executed on another chain.
The simplest working mechanism of such an atomic swap is the following:
1. Let we imagine that a we want to make an integration between a P public and C consortium network in a way that a Tc transaction is executed on the C consortium network if and only if a Tp public transaction is executed on the public network.
2. We create a random x secret value and a H(x) cryptographic hash of the random number.
3. We create a timed hash contract or transaction in a way that the transaction is on the blockchain but it will be executed if and only if x is openly presented on the chain, usually with the help of a transaction. If the x value is not presented in a certain time period, the transaction is automatically reverted. In case of Bitcoin for instance this revert means that the cryptocurrency is efficiently transferred back from the multisig wallet to the sender. However this is not necessarily the only possible use-case. Any logic with a transaction execution and revert can be used.
4. If x secret is presented on one system, the transaction will be executed. As in this case x secret is public, it can be used to execute to other transaction as well on the other system.
Certainly the system might have some disadvantages that can be fine-tuned: like in non smart contract based systems, the double spending should be avoided by the algorithm and the participants should be effectively online during the swap. On top, privacy might be an issue as well.