Where can I see my purchase history?

Open your profile and choose Activity. It has subtabs for sales and purchases, listings, offers, and Vault & transfers (vaulting, activation, and plain transfers). Each entry links to the transaction on the chain's block explorer.

Everything shown is read from the blockchain, so it may lag a confirmed transaction by a minute or two while the indexer catches up. The explorer is the authoritative record; see How do I find my transaction hash?.