It's just a standard and it makes things easier for everyone.
Why does it have to be 1 specifically?
It doesn't. It's just a standard. If you want so in your game, you can make all your vectors of length 2. I wouldn't call it normalized though because normalization has a mathematical definition of its own.
As long as you have all your vectors in the game standard (all of them of unit length N, doesn't matter what N is, but 1 is pretty standard), you could still do the direction*speed technique.
All that said, I don't know anyone who uses other unit lengths other than 1. I just want you to know that it doesn't have to be so, it's just much easier for everyone.
How would I use it?
A normalized vector would signify direction. That makes it convenient to mash up with speed.
For every frame, instead of doing
position = newPosition or
position = position + movementVector, you could generalize everything with
position = position + direction*speed.