Money needs to be divisible so that it can effectively act as a medium of exchange. If money is not divisible, it becomes hard to buy small items and it becomes difficult to have precise pricing.
Imagine, for example, if we tried to use cattle as money. This would be a big problem because there are surely many things that are not worth a whole cow. If you wanted to buy a movie ticket, for example, what you do? You couldn't cut off some percentage of the cow and give it to the ticket taker. If money is going to be useful, it has to be divisible so we can buy small things and so we can have precise prices (imagine trying to distinguish between something that cost $2 and something that cost $2.50 if you were using cattle as money...).
The reason money needs to be divisible is actually very simple and quite logical. To start with, divisibility is one of the four properties that makes a commodity, i.e., something of use or value, into exchangeable money.
Four Properties of Money
Durability: Maintains physical characteristics over time, like rocks or metals.
Transportability: Easily and conveniently taken with a person from one place to another and given from one person to another.
Noncounterfeitability: Cannot easily be duplicated by unauthorized persons, which is why Caesars in Rome (rulers everywhere) required their likenesses, i.e., their portraits, to be imprinted on the metal coin of the realm.
Divisibility: Readily made available in varying increments, e.g., (mixing American and English money values) halfpenny, penny, nickle, farthing, quarter, shilling, half-dollar, crown, in order to be easily exchangeable for items as small as a bread roll, as big as a breadbox, as large as a bread oven or as huge as a cargo ship--and a cargo of bread.
So, the reason money needs to be divisible is because in order for a commodity (something of use or value) to be used to buy and sell our daily, short-term or long-term needs and wants, like food, eyeshadow, aftershave, college tuition, or a chateau in the Swiss Alps, money needs to be readily and conveniently available in varying increments that are conveniently exchanged for any size or quantity of good or service. It has to be divisible into small enough increments to buy a single Tootsie Pop at the corner candy shop while also available in large enough increments to buy the 42-inch flat screen TV at the local Best Buy (or the Ferrari or Rolls-Royce or yacht you dream of).
For anything to act as the medium of exchange it should be available in a wide range of values so that all items, whether very expensive or extremely cheap, can be bought and sold. It is for this very reason that money should be divisible.
In the US, the smallest possible unit of available money is 1 penny and it is available in multiples of pennies (5, 10, 25, 100, 500, etc.). This enables buying something as cheap as a candy and as expensive as a house or ship.
Divisibility of currency is also the reason why metals such as gold, silver, bronze (not a metal, but an alloy of metals), etc. were so commonly used in human history as the currency. Compare this to the barter system, where live animals were used instead of money. How would you buy an item that is not worth an entire animal. Say 10 kg of grain is worth 1 sheep, but how do you pay for a candy? With a small piece of sheep?
Thus, we have a system of money/currency that is divisible into smaller units.