Why are we not moving 13 -> 9? It sends a piece back and is building on our wall. I understand that the recommended play gets pieces moved into scoring position, but sending the opponents checker back seems like a better play to my newb brain. Can someone explain?

There are more good things to do that just send checkers back. You want to think about how you plan to win the game. Are you playing an attacking game? Given that you only have one inner board point, that's not such a good idea.

You can also see that sending the second checker back gives your opponent more opportunities to get an anchor, which is bad for you.

The 20-point anchor is a huge win. Since your opponent has built an inner board point in front of you, it's very valuable to step over it.

Also, a big stack of checkers is generally bad. You're stacked up on the 6 point, and it's helpful to put those checkers to work building inner board points.

There are some other concepts, like connectedness between your front and back checkers, but those ideas tend to be harder for new players to conceptualize. The basic idea is that it can often be bad for you to leave your back checkers all alone. It makes it much more difficult for you to run them because there are no safe places to stop. So moving the back anchor forward and leaving the midpoint in place is helpful for the ultimate goal, which is to get your checkers home.

So it isn't that your move doesn't have benefits. It's just that there are better things that you can be doing.

I'm not an expert in either backgammon or chess, but discussions like this always make me compare the two games. In both, you always need to weigh tactics against strategy. Strategy covering long term development, and tactics covering short term actions, usually attacks. In backgammon a tactical advantage is usually short-lived (at least at this stage of the game) due to the randomness inherent in the game. More luck is needed to sustain a tactical edge into a decisive position.

Making the 20 and 4 points will always be advantageous, but the hitting play will only be successful with more good rolls right away.

XG++ (7-ply analysis) gives a slightly different answer
13/9* 6/4(2) comes out on top with 23/20(2) 6/4(2) -0.02 behind.
Not sure if the image is going to come out here..