I’ve always thought a geometric interpretation is a natural way to think about it. Part of multiplication can be described as “scaling”–if a multiply a number by 2, 3, or 4, what I’ve done is double, triple, or quadruple, respectively, that number’s distance from zero.
Multiplication by a negative number “reflects” that number across zero. What I mean is that if I start with, say, 3.5, and multiply it by -1, then my new number is still 3.5 units from zero, only on the opposite side from where it started. If I do that reflection twice, I’m back where I started.
So, breaking it down step by step, multiplying -2 * -3:
Start with 3, think of it as being three units to the “right” of zero. The “-” in -3 reflects it across zero, so now we’re three units to the “left” of zero. Now multiplying it by 2 doubles that distance, so we’re six units to the left of zero. The final “-” (in -2) flips it again, so we’re now six units to the right of zero: -2 * -3 = +6.
If you want to toss in complex numbers, I think it becomes even nicer to think of. A complex number has a real part and an imaginary part; picture a plane with two coordinate axes, the horizontal axis denoting the real part, the vertical axis denoting the imaginary part. This is the complex (Argand) plane, an extension of the usual real number line. For example, the complex number 4 - 5i would correspond to the point on the plane that is four units to the right, and five units down from the origin (= the complex number zero).
Multiplying in the complex plane has a very pretty picture. I can indicate any point in the plane by identifying two things: 1. How far it is from the origin, and 2. What angle does it make with the positive real axis, measured counterclockwise from the positive real axis.
If I take two points (complex numbers) in the plane, and want to multiply them, the result will be 1. Its distance from the origin is the product of the original numbers’ distances from the origin, and 2. Its angle (as described above) is the sum of the original numbers’ angles.
So, for example, if I take one number that is 3 units from the origin, at an angle of 37[sup]o[/sup], and another number that is 2.5 units from the origin, at an angle of 113[sup]o[/sup], the product of those twon numbers is 7.5 (=3*2.5) units from the origin, at an angle of 150[sup]o[/sup] (=37[sup]o[/sup] + 113[sup]o[/sup]).
The number -1 is one unit from the origin, and has an angle of 180[sup]o[/sup]. So multiplying by a negative number rotates the plane by 180[sup]o[/sup]–if I do this twice (a negative times a negative), I rotate the plane by 180[sup]o[/sup] + 180[sup]o[/sup] = 360[sup]o[/sup], and we’re back where we started.
I know this is kind of verbose, and that probably gets in the way of clarity. Don’t let that throw you, it’s my fault for not making this any clearer; a picture would make it clear that it’s really a simple idea; this site might be worth playing around with:
http://www.accesscom.com/~lillge/pgc/pgc-gallery2.shtml