We use a location/scale transformation of a base CDF which we show by $A$, to have its posterior which we show by $B$. $\alpha$ is the location parameter and $\beta$ is the scale parameter.
Let $X$ be the base random variable with mean zero. The transformed variable is $Y=\alpha + \beta X$.
Why is this true?
$$B_{\alpha,\beta}(Y)=A\left(\frac{Y-\alpha}{\beta}\right).$$
But I don't understand what allows us to use that standardized coefficient to connect prior $A$ to posterior $B$? Why does it depend on $Y$ and not $X$?
This looks like a simple location-scale change to Cumulative Distribution Functions (with $\beta>0$) rather than a prior-posterior change. You could write $$B(\alpha +\beta x)=A(x)$$ if you want this in terms of $x$. In more detail: $$B(y)=\mathbb P(Y \le y) = \mathbb P\left(\alpha +\beta X \le y\right) = \mathbb P\left( X \le \frac{y-\alpha}{\beta}\right) = A\left( \frac{y-\alpha}{\beta}\right)$$ and $$A(x)=\mathbb P(X \le x) = \mathbb P\left(\frac{Y-\alpha}{\beta} \le x\right) = \mathbb P\left( Y \le\alpha +\beta x \right) = B\left( \alpha +\beta x \right).$$