In the second line of the solution you added the exponent to both sides of the equation -- why? How is that the correct thing to do?

First rule of algebra: Whatever you do to one side of the equation, you have to do to the other.

Philosophy of tackling algebra problems: If you're solving for x, you want to get x on its own on one side of the equation. You generally do this by seeing what was done to x, and then undoing it.*

If you have something under a square root, try squaring it. If you have x + 3, try subtracting 3.

I think Greg and Greens explained the rest of it quite nicely.

*Note, there are other rules and special cases you have to watch out for, but it sounds like you're probably not there yet.