Exchanges interact with the Bitcoin network, and the blockchain, whenever people deposit on them, or withdraw BTC from them.
You can think of exchanges as services which are effectively like a bank account for multiple currencies. For example, they may offer a USD account and a BTC account. And they let you:
- Deposit or withdraw USD (using wire transfers or other means of sending fiat)
- Deposit or withdraw BTC (using Bitcoin transactions, on the blockchain, or possibly using Lightning)
- Sell BTC (which decreases your BTC account, but increases your USD account correspondingly)
- Buy BTC (which increases your BTC account, but decreases your USD account correspondingly).
Only the second of these actions involves any interaction with the Bitcoin network. The exchange operates a large Bitcoin wallet which holds all the funds of all its customers' BTC accounts, plus a database that remembers how much of it belongs to every customer (their account value). So this explains why trades don't incur on-chain transactions: they don't affect the exchange's wallet, only the database entries.