Children come to school and are taught about numbers - what we now call "natural numbers" - positive whole numbers. They learn to add them and multiply them by using demonstrations like "Johnny had 6 bananas" and "I have 5 bags of 4 apples". After they have understood this (or haven't), they go on extend this notion to fractions, with pieces of cake, and then to negative numbers, with things like debt.
The kids learn to add and subtract negative numbers, and that sort of makes sense. And even things like 3 * (-4) make sense as "three people owe me 4 dollars each". And then, the moment comes when the teacher announces, "Negative times negative is positive".
Why would she say such a thing? Here is a webpage that discusses why this would be true, through many different plausible explanations. But in fact, the real question is not "if we were to multiply two negative numbers with each other what would we get", but rather "why would we multiply two negative numbers in the first place?", or actually "Why even introduce them?" It is hard to believe this, but until around the 18th century negative numbers were not accepted by Mathematicians. They were called "imaginary numbers" (later on a new type of number started being called imaginary -- see Complex Numbers). Almost all of high school and college Mathematics, when developed, was done without any negative numbers at all! The calculus was developed without negative numbers! Cartesian geometry was done only in the first quadrant! Algebra was done only with positive numbers!
Debt might be a nice way to explain negative numbers, but instead of using negative "numbers" we could just use the word "debt" when needed and do calculations in that manner. We wouldn't need to consider them as numbers.
So, why were they introduced? It begins with Algebra. Imagine what a quadratic formula would be like with no negative numbers. We couldn't write the equation x2 + 2x = 8 in the form ax2 + bx + c = 0, since c would have to be -8, a non-existant number. It turns out, and this is how the original Algebra was developed in the book Kitab al-Jabr wa-l-Muqabala, that we would need six forms for the quadratic equation -- each with a different solution:
squares equal roots (ax2 = bx)
squares equal number (ax2 = c)
roots equal number (bx = c)
squares and roots equal number (ax2 + bx = c)
squares and number equal roots (ax2 + c = bx)
roots and number equal squares (bx + c = ax2)
In addition to the many different forms, each solution was truly different (if you know how we solve the modern version of the quadratic formula, try to use the same ideas to solve these without negative numbers). The introduction of negative numbers and their rule of multiplication, first given by Bhahmagupta, simplified the quadratic equation into the form we use today, given by Bhaskara Acharya. Many other types of situation with many cases were united into one form. So it seems that negative numbers and their rules of multiplication help us see many different cases as one -- it helps unify Mathematics.