How does this Boolean simplification work?
It's just an identity. It's true. You can do that straight away.
A + AB = A(1 + B)
More generally:
AC + AB = A(C+B)
In this case, the boolean algebra identity is similar to what you usually do in normal algebra.
I don't think there is any proof for this except you can verify this with a truth table.
And I think there is a simpler and more intuitive way to perform this simplification in one step instead of three. If you consider AB, it can only be true if A is true. If A is false, then AB is also false. So if A is true, then A + AB is obviously true. And if A is false, then A + AB is also false.
So with this understanding, we can directly say A + AB = A