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I know well that I can sell and buy cryptocurrencies like BTC and ETH on exchanges like Coinbase, I do it almost every week.
But my question is: how does an exchange materially convert dollars or euros to bitcoins? If I were the exchange and a user gives me one hundred dollars and tells me to "convert them to bitcoins"; what is it that I as the exchange then do?
You find some other people who want to exchange Bitcoin for dollars.
It's no different to exchanges that exchange dollars for euros.
Exchanges are platforms for users to trade currencies with other users.
The usual flow is that users deposits dollars, euros, bitcoins, or whatever into their account on an exchange, and then use their account balance to offer a trade for whatever they want to acquire. The exchange merely offers the platform, matches and executes the trades, and facilitates deposits and withdrawals. The exchange itself is not participating in the trades. Since users can only trade whatever balance they have in their accounts, the exchange also guarantees that the counterparty has the funds to participate in the offered trade.
There is another distinct business model, called a brokerage, where users directly purchase from the platform. In that instance, the brokerage is the counterparty for all trades and must acquire stock to sell, e.g. from miners minting fresh bitcoins, exchanges selling off the trading fees they collected, or market makers with large holdings.
