Hi runjoh,
Sorry for the late reply. I'm glad you have the info you need to understand the circuit because frankly, after watching the video sevral times, feeling like a hostage, and drawing a schematic from it, I have no idea (I have a great liking for the 555 and admiration for its inventor and have done many different things with it for about 8 years, btw).
Perhaps I have misunderstood something in that person's video and there is a mistake in my schematic, but I will say, along with Audioguru, but in different words, the way the 555 is set up reminds me of dodgy 'experts' who share their circuits that as a beginner I couldn't tell the difference between 'good' and 'bad'/'sloppy' and 'BS' circuits by people desperate for attention but lacking scruples, and there are a couple of warning signs in the circuit here: There are no bypassing capacitors on the 555 supply pin; there is no 100nF capacitor from pin 5 to ground, and it is left floating (pin 7 is also left floating, but that is not such a terrible thing, except for the fact that, generally speaking, floating IC pins ARE A BAD THING); IMO there is a serious lack of resistors in a few places - PNP current-limiting base resistor, an NPN pull-down is missing, there are no current-limiting resistors on the LEDs (3.7V - 1.7V = 2V; 2V/0 Ohms = the only current-limiting for the LEDs is the 555 output resistance); a 47 uF capacitor...? - Seems unnecessarily huge capacitance to me (my personal rule-of-thumb is to - where possible - use as small a capacitor as possible with a larger resistor, not a small resistor and a huge capacitor). Also, I hope I have copied the circuit wrong because if I haven't, the NPN emitter connection is just weird and looks like a bad joke.
A quick simulation with a voltage source to replace TSOP1738 didn't tell me anything as what was most memorable was that there was an 8mV output voltage from the 555 pin 3... I assume I missed something in the instructions and my schematic is different to what the person says in the video, but I sat through it some four times to check I had copied it according to the instructions...
By the way, what's that from pin 6 to pin 2 on the timer - an unmarked 0 Ohm resistor, or what? The guy doesn't even say what it is.
Schematic (I am very open to corrections from anyone who can be bothered to watch the video):
I guess you understand that:
1) Press a (/any?) remote control button.
2) TSOP1738 output goes low > BC557 turns on > BC547 turns on.
3) From there, I wouldn't like to guess because of the BC547 emitter connection that is a new concept for me. All my brain sees is ~1.85V at the emitter and ~3.7V at the collector of the BC547, from there my brain melts.
The 555 trigger pin needs to go low (1/3 Vsupply) for the output to go high and the output will go low again when the capacitor charges to 2/3 Vsupply. I don't understand why the timer is biased at Vsupply/2, it's neither one thing nor the other I would have expected.
If that really is a valid bistable circuit, I'll bet I'm not the only person who would use it instead of e.g. a flip-flop if a CD4013 were unavailable. As you say it works well, so be it.
Anyone who understands this circuit, I'm all ears and humility.