Cryptocurrency exchanges are privately-owned platform platforms which facilitate the trading of cryptocurrency and convert fiat currency (USD, EUR etc.) to crypto (Bitcoin, Ethereum, etc.), and vice versa.
It is a marketplace where buyers and sellers come together to trade assets at specific prices. It exists as a platform on which market participants transact without having to search for a buyer or seller willing to trade with them. The exchange takes on this task.
Trading through an exchange is highly preferable for traders since a large number of users are gathered in one place, which generally allows for more liquidity (i.e., the availability of an asset in the market) and, theoretically, more competitive prices.