There are a couple of ways to look at this. I think about it as solving a puzzle (not a very complicated one). The equation says, "I'm thinking of a number that when you square it, you get 9. What could the number be?" The answer to that is that $x =3$ or $x = -3$ - those are the only answers, which is exactly what the equation $x = \pm 3$ means.
You can also think about it that putting the $\pm$ on both sides is redundant - the equation $\pm x = \pm 3$ contains 4 different possibilities: $x = 3$, $x = -3$, $-x = 3$, or $-x = -3$. But those possibilities only lead to two solutions: $x=3$ and $x-3$. So the other two equations are redundant.