Man that sure seems like a violent ride.

Can someone who understands this explain something to me? He appeared to reel that yellow car right in---but then couldn't pass him. Why? Was the other car slower but had better brakes, so he could stuff him into each corner? Or did the other driver get faster when he realized he'd been caught?
Or something else?

kept looking for the pass and it never happened.

You can go way faster when you aren't fighting wind resistance, the closer you get to someone the faster you can accelerate towards them, but the moment you try passing, you get the full force of the wind on you, and you lose speed. So could have been the other cart had more power, or could be the driver filming this was bigger/heavier.

I don't really think it's a matter of wind resistance. Drafting with a car behind a semi offers measurable fuel benefits out to near 100 ft, but these things are so much smaller than a semi that it's likely less than 10. Most of the time in the video (especially when we're seeing quick gains) the following kart is much farther back than just a few feet...and in several cases, he's off to the side as well.

I'd put the difficulty of passing down to twisting, narrow roads and an aggressive lead driver.

It could be. Wind resistance is why geese fly in formation. They take turns being the lead so the entire flock benefits.

OBVIOUSLY, the guy in the lead used a mushroom.

It could be, although it's exceedingly difficult to find any meaningful numbers for distances that aren't terribly vague, and what you do find is almost always for something other than karts (semis, F1 racecars and bikes).

I still think that the open, small, only semi-streamlined form of the karts would tend to make the effective distance less than what we saw in most of this video. The lower speeds than F1 cars would tend to improve the slingshot tactic, since the effect of wind resistance increases with the cube of velocity. Which again, doesn't seem to explain the drastic slowdowns when they're not at top speed anyway.

We need someone involved in this hobby to weigh in and give an expert opinion.

It's wind resistance, it makes a massive difference. When they're really far apart the rear driver is just driving better and shaving time off the lead, so he catches up, but once there it's down to drafting to get the little burst of speed to get alongside. The carts are probably approximately equal in power, so he reaches level from the draft position easily enough, but can't keep the momentum to get a lead with the new air resistance on him, just draw level. So they're level, but obviously there's only one sweet racing line to take to keep your speed up and lap time down. You can either pass on the outside (in which case you have to go faster into the turn to stay ahead) or the inside (in which case you have to turn sharper at speed to stay ahead) both of which are risky, or you can return safely to the racing line - i.e. not by swerving into him, but by conceding the lead to him and dropping in behind him. If you do that, you take less risk and give yourself the chance to try again because you're in draft position again. He needs to stay as close as he can and find the right place to overtake so that his superior driving can give him the lead into the racing line of the next corner, at which point he gets right of way and the position advantage the lead gives. Sometimes that's not even possible and lead to what some would call boring races (Monaco Grand Prix) where the leader is decided on the first corner and doesn't change unless they crash out.

I'll draw two parallels:
1. DRS in F1 racing, where a tiny part of the tail opens up for a small part of the track, which drastically increases speed and allows for more interesting races because it almost ensures overtaking. You can also see the same application of the racing line and people conceding position or trying to take different lines and spinning out or locking up.
2. In cycling, the commonly quoted figure is that you can save 40% of your energy by drafting behind a leading cyclist. The Tour de France and every other cycling road race is defined by drafting, cos no lone cyclist would ever be able to keep pace with the peloton which 'cycles' riders in and out of the wind-protected bunch throughout a day. This should convince you more as cyclists are not streamlined objects but still offer significant gains. Go and watch a cycling sprint finish - it's a case of whoever gets behind the fastest guy wins by conserving energy in his wake until it's time to burst out alongside and pass.