https://arxiv.org/abs/1405.7907
In particular I would like to understand this paragraph, right before chapter 6:
This route to the Born Rule has a simple physical interpretation. Take the wave function and write it as a sum over orthonormal basis vectors with equal amplitudes for each term in the sum (so that many terms may contribute to a single branch). Then the Born Rule is simply a matter of counting – every term in that sum contributes an equal probability.
It helps to see what is going on if we simplify the notation by leaving out the environment and observer.
If you have a state representing the superposition of two outcomes with unequal amplitudes, like:
$\sqrt{a}\left|x\right>+\sqrt{b}\left|y\right>$
then we want to rewrite it as a sum of orthogonal equal-amplitude terms:
$\left(\left|x\right>_1+\left|x\right>_2+\ldots+\left|x\right>_a\right)+\left(\left|y\right>_1+\left|y\right>_2+\ldots+\left|y\right>_b\right)$
where there are $a$ terms with an $\left|x\right>$ as part of the state and $b$ terms with a $\left|y\right>$. By Pythagoras theorem in $n$ dimensions, the length of a sum of $n$ mutually-orthogonal unit vectors is $\sqrt{n}$, so this would match our original expression. Then we can simply count branches to find the probability. There are $a+b$ independent branches in total, of which $a$ give an $\left|x\right>$ and $b$ give a $\left|y\right>$.
The trick is to make those apparently identical outcome states orthogonal. That is done by combining each with a different environment-observer basis state.
