I looked this up a while ago (on my own personal hunt for why we use money), and I read something along the lines of "Farmer Jim wants two cows. Farmer Alice is only selling cows for two warthogs a piece. Farmer Jim lives in a place where warthogs don't live, so he can't buy the cows. But, with money, instead of Farmer Jim looking around the world for warthogs, he can just give Farmer Alice 200$ and get the cow."
Basically, itwas saying that when we traded, you had to have what the person wanted to trade you for, but once we got money, you were able to, I don't know, be more flexible? Have more ways to get that money? I don't exactly remember, but that's the gist of it.
Anyway, yes, I know this is a rather broad question, but I'm hoping someone can give me a little insight.
Edit: title typo. My apologies.
Look at David graeber's Debt, which challenges the idea that barter was the predominant form of exchange before coinage.
The first use of coins was seen in Lydia, c. 600 BC. I don't know why the Lydians switched to coins, but I will give you a reason why the Greeks switched to this system.
Coins were used in ancient Greece to facilitate trade and make it easier. Multiple cities issued coins, printing their own emblems on them to signify "city ID" so it really came down to which cities were trustworthy and which ones were not; could you really take the given weight at face value? For instance, you're trading somebody who hands you a silver coin supposedly weighing 5g (completely arbitrary number). As a merchant, could you accept that it was actually 5g or would you have to weigh it to make sure? Athens issued coins and it was known to be extremely trustworthy; if I were a merchant receiving an Athenian coin, I would trust it. Trading would be easier and go much smoother in general. If it were from a lesser-known city or one with a questionable reputation, I would be inclined to weigh it for myself instead of trusting the supposed value.
Source: "The Ancient Mediterranean World: From the Stone Age to A.D. 600" by Robin Winks
People continued to barter even after coins were developed. People used non-paper, non-coin substances as currency before the development of paper and coin currency, like olive oil in the ancient Mediterranean.