The ancient Mesopotamians had a system of money because money is very convenient. It is very difficult to run a large civilization without using money.
Without money, people must resort to barter. Barter is very inconvenient for a number of reasons. The most important of these reasons is that people need to have what economists call a double coincidence of wants in order to successfully engage in barter. Let us imagine that we are in a world where there is no money. I have the ability to read and write and I want two chickens for dinner tonight. In order to barter, I will need to find someone who has two chickens and who needs something read or written. If I cannot find this, I will have to find out what the person with the chickens wants and then I will have to go find someone who will trade me those things in exchange for some reading or writing. This is very inconvenient. When a society has a system of barter, its economy will not be very efficient. People will waste time trying to find other people to trade with.
There are other factors that make money more convenient to use than barter. Money is good because you can divide it up however you want to buy small things. It is good because it will not spoil or die, which means that it will still be valuable tomorrow or next month. It is good because everyone knows how much it is worth. All of these things make money very useful. The ancient Mesopotamians created a system of money because it had become clear to them that having money was easier for them and better for their economy.