Why is negative times negative = positive?

This is pretty soft, but I saw an analogy online to explain this once.

If you film a man running forwards ($+$) and then play the film forward ($+$) he is still running forward ($+$). If you play the film backward ($-$) he appears to be running backwards ($-$) so the result of multiplying a positive and a negative is negative. Same goes for if you film a man running backwards ($-$) and play it normally ($+$) he appears to be still running backwards ($-$). Now, if you film a man running backwards ($-$) and play it backwards ($-$) he appears to be running forward ($+$). The level to which you speed up the rewind doesn't matter ($-3x$ or $-4x$) these results hold true. $$\text{backward} \times \text{backward} = \text{forward}$$ $$ \text{negative} \times \text{negative} = \text{positive}$$ It's not perfect, but it introduces the notion of the number line having directions at least.

Informal justification of $\text{positive} \times \text{negative} = \text{negative}$

Continue the pattern:

$$ \begin{array}{r} 2 & \times & 3 & = & 6\\ 2 & \times & 2 & = & 4\\ 2 & \times & 1 & = & 2\\ 2 & \times & 0 & = & 0\\ 2 & \times & -1 & = & ? & (\text{Answer} = -2 )\\ 2 & \times & -2 & = & ? & (\text{Answer} = -4 )\\ 2 & \times & -3 & = & ? & (\text{Answer} = -6 )\\ \end{array} $$

The number on the right-hand side keeps decreasing by 2.

Informal justification of $\text{negative} \times \text{negative} = \text{positive}$

Continue the pattern:

$$ \begin{array}{r} 2 & \times & -3 & = & -6\\ 1 & \times & -3 & = & -3\\ 0 & \times & -3 & = & 0\\ -1 & \times & -3 & = & ? & (\text{Answer} = 3 )\\ -2 & \times & -3 & = & ? & (\text{Answer} = 6 )\\ -3 & \times & -3 & = & ? & (\text{Answer} = 9 )\\ \end{array} $$

The number on the right-hand side keeps increasing by 3.