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).   

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...).