Normalized means the magnitude of the vector is always 1 and the vector then only gives you the direction. This allows you to, for example, multiply it with a speed value of your choice and always know the speed will be the same going in the direction of the vector.

So when you before hade vector forward and right, you can simplify and say it got twice the magnitude and therefore went faster. When you normalize them you get the correct direction, but always with the length of 1.