The foreign exchange market is a huge over-the-counter market, primarily intermediated by large dealer banks with subsidiaries around the world (sometimes referred to as “money center” banks).
A buyer or seller of a currency is almost certainly at or trading with one of these banks. The dealer banks make money via a “spread,” where every buyer or seller of a currency effectively pays the dealer they’re trading with. For example, if 1 cur1 = 1 cur2, a buyer of cur2 might have to pay 1.01 cur1 for it, and a seller of cur2 might receive 0.99 cur1 for it, with the dealer taking .01 on both sides.
The forex market is large and highly competitive, and balance between the prices of any two currencies is just determined by supply and demand— as people trade cur1 for cur2, the supply of cur1 at dealer banks will rise, and the supply of cur2 will fall, so the price of cur2 will rise relative to cur1. From a practical standpoint, as more people trade cur1 for cur2, a dealer bank will have to replenish its reserves of cur2, which it will often do by borrowing it.
In my opinion, the simplest way to think about all this is to remember that the primary source of demand for any currency is people who want the stuff denominated in that currency (be it consumer goods, financial or real assets, etc.). When more people want that stuff, relative to the stuff in other currency areas, the price of that stuff (i.e., the value of the currency) rises. For example, Canada is a huge oil exporter, and if you want Canadian oil, you first have to get Canadian dollars. As a result, when demand for oil rises, the value of the Canadian dollar rises.