Zero is a pretty fascinating concept in terms of mathematical development.
Most importantly, for math, it gives us a systematic characterization of "nothing." Now, this may not seem important, until you realize that without this characterization of 0, negative numbers can't be formed (see link below). This is especially important in the area of logic. If we do not have zero, we will simply be "waving our hands" instead of constructing systematic, indisputable proofs whenever zero would be involved! You can't use undefined terms in a real proof. It would be like making laws that address hornswogglers without saying what a hornswoggler is!
It seems that the original problem in defining zero as a number was how most math in ancient times was conrete and practical. Either I have some quantity of sticks, or I have no sticks. I didn't really need to think of the fact that I had "zero" sticks. I just needed to know my stick supply wasn't there anymore. The problem came about if I wanted to name certain numbers or fractional numbers of sticks. If I had a lot of sticks, I needed a new symbol for each new place-value because I had no place-holder. Think Roman Numerals for this issue. Zero as a place holder is especially important as we see that there is no way we could have expressed small decimals or very large numbers without it!
I hope this helps with the explanation.