IOTA Smart Contracts work as a layer 2 (L2 for short) extension of the IOTA Multi-Asset Ledger (Layer 1, or L1 for short, also sometimes called the UTXO Ledger).
In IOTA Smart Contracts, each L2 chain has its own state and smart contracts that cause this state to change. As validator nodes execute the smart contracts, they tally these state changes and write them into the chain. Each time they update the state, they collectively agree on a new state and commit to it by publishing its hash to L1.
Each Layer 2 chain is functionally similar to a traditional blockchain. However, ISC chains can communicate with Layer 1 and each other, making ISC a more sophisticated protocol.
IOTA Smart Contacts multichain architecture.
You can find the comprehensive overview of architectural design decisions of IOTA Smart Contracts in the ISC white paper.