Don't understand the play here... splitting here is very counter-intuitive to me; seems like it's just begging to be gammoned for what? 13/8 13/9 seems to provide safety whilst bringing in more ammo to attack the lone checker. What am I missing?

13/8 13/9 doesn't bring more ammo to attack White's back checker (at least not immediately). The spare on the 8 point doesn't really improve the attack (because in a situation where you hit on the 2 point, the fact that the 8 point gets broken is close to irrelevant) and the checker on the 9 point is out of range. What 13/8 13/9 does do immediately is make it much easier for the back checker to escape. Any 5 roll except 5-1 puts him in the clear. As it stands, with the midpoint there, 6-5 is blocked and 5-2, 5-3 and 5-4 all leave a direct shot on the checker as it tries to escape. Because you probably won't be in a position to put White on the bar next turn, you leave him at least two shots to roll 5-x and escape. If he does roll that, your game is essentially going to be over because you don't have the timing for a holding game. Your board will get crunched before White leaves a shot. The midpoint is serving a few other functions as well, primarily making it harder for White to move his own midpoint. For instance, after 13/8 13/9, 6-1 is a great roll for White. As things stand (or after 24/15) it's bad - White is forced to leave a shot.

24/15 is not as dangerous as it looks because White doesn't own the 5 point and is not really in a position to close you out. Certainly if he whiffs the outfield hit, your back checker is in zero immediate danger (of either getting blitzed or primed). There's a modest increase in gammons but nothing too bad. Note also that in addition to not breaking the midpoint, 24/15 increases outfield control. 5-4 and 5-2 become worse rolls (5-2 in particular is no fun at all) because running the back checker now leaves a double direct shot. (5-3 will be used to hit).

Wow, thanks for the very thorough analysis! There's a lot going on in this position that I hadn't seen; hopefully I'll be quicker to spot those issues next time!

24/15 works on escaping your back checkers while keeping the rest of your position intact. The play is not particularly dangerous since white has an open 5 point and few attacking numbers. If you do get hit, you'll usually have return shots and you also have a strong board. As Chris pointed out, breaking the midpoint takes pressure off white's midpoint, allows him to escape his back man more easily, and doesn't provide much additional attacking potential if white fails to roll a 5. It also makes it more difficult for you to escape your own back men.

If you consider the play risky, imagine if you break your midpoint and then white escapes his lone back checker. You're now left with 2 men back who are completely disconnected from the rest of your checkers, and white having full outfield control. Now the risk associated with any play which tries to free your back men (which has to be done eventually) has dramatically risen. In fact, white will often have an efficient double in this variation.