I guess it was like this: After the invention of natural numbers, one day someone (let us say $A$) was collecting apples. When returning home, he met one of his friends, $B$, who was quite hungry and had nothing to eat. $A$ decided to give to his friend three apples, and $B$, full of thankfulness, said that he would give the three apples back to him as soon as possible (by the way, this was also the birth of ASAP). And $B$ told to himself "I owe my friend $A$ three apples, do not forget it". The number $-3$ was born.
The good thing about the existence of negative numbers and 0 is that renders the operation of substracting much more coherent, in the following sense: Think of two natural numbers $a$, $b$. If $a>b$, then we can think of taking $b$ out of $a$, so that we obtain the quantity $a-b$. But we cannot take out $a$ from $b$! This is quite annoying. But negative numbers allow us to say: I take $a$ from $b$ and there is a pending balance of $a-b$ units, which we write as $-(a-b)$. And we all know this is especially useful when talking about money... Hence, with negative numbers (and zero) two natural numbers can always be substracted, and computations work much more smoothly.
So, we have this new family of numbers comprising the naturals, zero and the negatives of the natural numbers. This collection of numbers is usually called "integers" and represented with the letter $\mathbb Z$. And this apparently innocent looking set is full of mysteries...
