Two that come to mind are Coinbase & Circle. I did a quick google search, possibly wrong terms. The information I got was useless. I am sure this has been discussed. If there is a good article/link on this it would be appreciated.
I think they got it from many ways. before the did this exchanges business, they need financial capital that probably they got from the seller or from their job incomes, etc.
so when they have this first financial capital, they can open exchange service. so they exchange crypto to fiat or the opposite.
Another way they can do it also is by borrowing from financial institution if they are able to prove to them they the business they are going into will pay off. Once a bank understands that there is going to be high return on investment, they will not hesitate to give them financial aid.