Zero product property

The statement

If $ab = 0$, then $a = 0$ or $b = 0$, or both.

Why zero is the only number this works for

Multiplying can land on zero only if something you multiplied was already zero. Nothing else gets you there — $7 \times 0.0001$ is very small, but it is not zero.

No other target has this property. If $ab = 12$, then $a$ could be $1$, $2$, $3$, $0.5$, $-4$ and endlessly more. Knowing the product tells you almost nothing about the factors.

What it means when solving

This is why an equation must be rearranged so one side is zero before you factor. From

$$(x-2)(x-3) = 0$$

you may conclude $x = 2$ or $x = 3$. From $(x-2)(x-3) = 12$ you may conclude nothing, and writing $x - 2 = 12$ applies a rule that does not exist. See solving by factoring.