High Order Butterworth Filter Math
So how did they go from the polynomial to the gain?

They didn't go from the polynomial directly to the gain. Your reference link actually omit something, then come from other place.

The equation 3 - A = 1 is related to the stability of the filter. It actually applies to second-order Equal-Component KRC filter (An implementation of Sallen-Key filter, with equal resistors and equal capacitors). For this type of filter the 

$$
Q = \frac{1}{3-K} \quad\quad\quad\quad\quad\quad\quad(1)
$$

\$K\$ is the DC gain of the filter, you see if K is equal to or greater than 3, the Q will be infinite, event be negative! This will make the filter unstable. So, K must be less than 3. That is, the system should have DC gain less then 3.

So the author intends to use this type of filter structure to make a Butterworth filter, and forgets to mention this detail. 

In a second-order system, the damping factor has a relation with Q

$$
Q = \frac{1}{2\zeta}\quad\quad\quad\quad\quad\quad\quad\quad(2)
$$

So, combine (1) and (2)

$$
K=3-2\zeta
$$

As we known, \$\zeta\$ should be between 0 ~ 1. The author choose 3 - A = 1, he actually set the \$\zeta\$ to 0.5, and then the gain should equal to 2. Because he choose the equal-component KRC implementation, so \$R_{f} = R_{1}\$, and the gain = 1+Rf/R1, just equal to 2. Then all are perfect.