Why is $-a \times -b = ab$? This might be a bit of a meta-mathematical question (if there is such a thing as meta-mathematics) but it confuses me slightly. 

I think of multiplication (whether rightly or wrongly) as recursive adding. For example, $2 \times 4$ is just adding 2 four times i.e $+2 +2 +2 +2 = 8$. 

I think of multiplying negative numerals e.g $2 \times -4$ as subtracting 2 four times i.e $-2 -2 -2 -2 = -8$. Unfortunately this analogy seems to break down when multiplying to negative numbers e.g $-2 \times -4$. I don't understand how $-2 \times -4$ can result in $+8$. 

What is the reason behind this, if any? Is my analogy incorrect? Is there a flaw in my fundamental understanding of multiplication?
Receiving 4 gifts of 5 dollars gives you a result of $4 \times 5 = +20$ dollars. This means you benefit by 20 dollars.

"Receiving" $-4$ gifts of 5 dollars means giving four gifts of 5 dollars. This results in a loss to you of 20 dollars, so $(-4) \times 5 = -20$.

Receiving 4 "gifts" of $-5$ dollars means someone is making you pay them 5 dollars, and doing it four times. That gives you a result of $4 \times (-5) = -20$ dollars.

"Receiving" $-4$ "gifts" of $-5$ dollars again means giving someone four of these (undesirable) gifts. Thus you are charging someone 5 dollars, and doing it four times. The result is of course good for you (if you don't care about karma), since you benefit by 20 dollars. So $(-4) \times (-5) = 20$. 

Here is a second approach. 

$4 \times (-5)$ is obtained by adding four copies of $-5$:
$$4 \times (-5) = (-5) + (-5) + (-5) + (-5) = -20.$$

So it is natural that $4 \times (-5)$ should be -20, by the "repeated addition" understanding of multiplication that you mentioned. Of course, we can't add "-4 copies" of -5 together, but pay attention to the pattern in the following list.

$4 \times (-5) = -20$

$3 \times (-5) = -15$

$2 \times (-5) = -10$

$1 \times (-5) = -5$

$0 \times (-5) = 0$

You will surely agree that if we are to multiply negative numbers at all, then the pattern should continue this way:

$(-1) \times (-5) = 5$

$(-2) \times (-5) = 10$

$(-3) \times (-5) = 15$

etc.
If you find natural that $2 \times (-4) = (-4) \times 2 = (-4) + (-4)$ you would agree that $a \times (-b) = - (a\times b)$. This can be made true even if $a&lt;0$ and/or $b&lt;0$ resulting in $(-a)\times(-b) = - ((-a)\times b) = -(-(a \times b)) = a \times b$.