The point is, what we do to the equation can affect the solution, and we sometimes have to account for those changes.
For instance, x=3 is a simple equation that has a simple solution. There is only one value for x that makes it true, and that value is 3. If we were to square both sides, we get which has two solutions, 3 and -3. By squaring both sides, we've introduced the possibility of a solution that wasn't a solution to the original. Similarly, when we take the square root of both sides, we have to remember to keep both solutions.
That's what we mean by the plus/minus sign in