Most crypto exchanges have some sort of buffer address (unique for every person) where you deposit your bitcoin and then this is sent over to their main wallet and your account is credited with the amount you deposited. How does this work?
Generally, the exchange will use a full node that is continuously checking the network for received transactions. Once it detects a transaction has been made to your deposit address it waits for the transaction to be confirmed, then it will credit you the balance. This data is stored in a private database managed by the exchange where they keep track of each user's balance. The bitcoins you deposited are then either pooled to pay for someone else's withdrawals or sent to a cold wallet (an air-gapped device that is not connected to internet) where they are stored securely if they're not going to be used in the near future.