- Currencies are indeed "actually" traded basically on what is called the "Interbank Market"
Here's a quick read on that:
https://en.wikipedia.org/wiki/Foreign_exchange_market
Let us compare and contrast!
Stocks (like AAPL) are (as a rule) traded on the two or three huge "stock exchanges" such as "Nasdaq", which everyone has heard of.
Commodities like gold and oil are (as a rule) traded on the two or three huge "commodities exchanges", such as "Nymex", "CME", etc, which many have heard of.
And to your question, currencies generally are traded on the huge "Interbank Market" (as nicely described in that article).
This is a (fairly! - see below) free market. The price is determined exactly, each second, by buyers and sellers.
It is an OTC distributed system, and very much an insider's market, which is broken in to tiers with the top tier being a small number of he world's huge banks.
The literal trading amongst this group (that is to say, "the database, communications, etc") is handled by a shadowy concern called EBS (which indeed, "they" set up) (again, you need only turn to Wiki .. https://en.wikipedia.org/wiki/Electronic_Broking_Services )
That is the only "real" price of a currency - the volume of the top tier is just overwhelmingly, astoundingly, large, and hence "is" the current price.
Just as Nasdaq continually reports the "actual" price of AAPL to the internet, and Nymex continually reports the "actual" price of gold and oil to the internet, the Interbank Market (on a literal technical level, basically "EBS") continually reports the "actual" price of EUR, AUD, etc to the internet.
So in a sense the simple answer to your question ..
"Where do currency prices come from?"
is:
"From a little-known behind-the-scenes technical platform called EBS. They handle the OTC trading of a tight group of the biggest players - the world's hugest financial institutions. They trade freely and openly in unbelievably large volumes, but each tier of access is very much a "club". {Indeed, some see it as a cartel; there was indeed a spectacularly large scandal about this in recent times.} This is all in general known as "the Interbank Market".
- When you go to a local exchanger at the airport, they simply use that current rate, i.e. ultimately from the "Interbank Market" system.
Your question is astute...
When "you or I" change some money, the money changer retail shop on the street is not "trading", it's not a situation where two parties offer bids/asks. The shop simply offers a fixed price to buy or sell different currencies, and that fixed price is simply based on the current price seen on the "actual" market. (Exactly the same thing happen when you buy some ounces of gold from a retail gold dealer.)
To make a contrasting example - when you buy a house from some person, it's an actual trade with bids/asks by the parties involved, which will settle at some price. And when you buy stocks (even 1 share) it's more or less an actual bids/asks situation.
Just to reiterate, this wiki article https://en.wikipedia.org/wiki/Foreign_exchange_market pretty much fully explains the interbank market. Which is indeed the worldwide currency "exchange" - which is analogous to the Nasdaq "exchange" for stocks or Nymex "exchange" for gold and oil.
Many see this as a cartel; as it explains on wiki:
"Unlike a stock market, the foreign exchange market is divided into levels of access. At the top is the interbank foreign exchange market, which is made up of the largest commercial banks and securities dealers..."
(indeed that precisely is where the price the OP is asking about comes from, via, on simply a technical level, the "EBS" platform which they set up and now use)
...which led to the spectacular "Forex Scandal" of our times, and here's the latest on that. Reading the articles on the zillion-dollar scandal, illuminates the overall scene.
(The only thing more "fixed" is the London gold fix, which is set each afternoon by five particular players in cahoots.)