Nothing will happen "under the hood" in the blockchain of a cryptocurrency when you trade on an exchange. The exchange has many wallets where the funds of the user are stored. The exchange has also a database where they store the information about the assets of the user, their trades and their resulting balance. Example:
A person A deposits 1 btc to exchange XY. Person B deposits 10 eth to exchange XY (these are transactions which you can see in the public blockchain of the respective blockchain).
Now the exchange has 1 btc and 10 eth and in their database they write analogous: "A has 1 btc; B has 10 eth".
Now person A wants to exchange his 1 btc into eth. So he buys 10 eth from person B and pays with 1 btc. This trade will not cause any transaction which will be added to the blockchain, because it is an exchange-specific process. When the trade is executed, exchange XY write in his database something like "A has 10 eth; B has 1 btc". (When person A, for example, now wants to withdraw his 10 eth to his ethereum-cold-wallet this would cause a transaction in the ethereum-blockchain.)