How did money become the medium of exchange?

Because of its universal acceptance.

Barter--exchanging goods or services for other goods or services--existed before money and it's still used. There are a couple problems with it...the first is the person who has what you need might not need what you have. If you're a barber who needs scissors, you might have to trade haircuts to ten people for potatoes. You'd then trade the potatoes for hams, the hams for beer and then the beer for scissors. You'd have to hire a secretary just to trade all the stuff people gave you for haircuts. The other problem is fractional worth. You make wagon wheels. A wagon wheel is worth more than a haircut, and you can't trade part of a wagon wheel for anything--they only work when intact. To get a haircut you must first sell a wheel for a barrel of crackers, which you can then use to buy haircuts, grease or whatever you need.

Money is better than barter first because everyone accepts it in trade for their wares, and second because you can make change for it. If you sell a wagon wheel for $10, and a haircut costs a quarter, you can take one dollar to the barber, get your hair cut and receive three quarters in change.