The range is the set of possible y values. A y is in the range if there's some x that can produce it.
So you're finding an explicit rule for the x that can produce a given y. If you have such a rule, you know the y's that can be produced.
Let's take a simple example, like y = x\^2 + 6, a parabola that we know starts at y = 6 and goes upward. We'll restrict it to x >= 0 so it's invertible.
What x produces a given y? Well inverting it, we get x = sqrt(y - 6). If y >= 6, we can use this formula to explicitly find the x that produces that y, that has the property that y = x\^2 + 6. Since that formula produces an answer for any y >= 6, we know that all y >= 6 are in the range.
But if y < 6, the formula fails. There is no x such that x = sqrt(y - 6). And that means there's no x such that y = x\^2 + 6 that will produce a y < 6. So the values < 6 are not in the range.