Trading happens in the venue's ledger, which is why it is instant. Money only crosses the boundary twice: when it arrives, and when it leaves. Both crossings are transactions on a public chain, and both are shown here with a link to an explorer that has nothing to do with us.
Tokens are held in the venue's vault contract, and the vault pays a withdrawal only when two signatures check out: the venue's, and your own wallet's, for that exact amount and destination. A voucher a compromised server wrote on its own would pay nobody. Where the venue holds bitcoin, deposit addresses are derived from a public key and withdrawals are signed on a device never connected to the exchange. The vault's balance is public, and the audit page lets you check what is held against what is owed.
Asking the venue what it holds…