If I understand your question, the notch in the track allows each wheel axle to become the "lever" (by analogy to a typical electrical switch) which completes the circuit to the LED. It's hard to see from the video where he physically mounted the LEDs and he may have placed them in or under the gap created by a notch. You might try asking him on youtube.
To your point, by notching the track you are breaking the electrical connection in the outer rail. But the two outer rails are electrically connected elsewhere on your layout - probably in multiple places, at lock-ons, etc.. From the video, he says he did this on one 10" section of track as opposed to on his entire layout. So IF the notches materially affected the performance he could have added wires (shown in purple) underneath his bed to provide outer rail continuity to the broken lower rail. As suggested above, there are other methods which don't require notching - I'm simply taking your original question at face value and speculating on how the youtube version works (and quite well I think!).
What kind of track do you have? One nice thing about 3mm LEDs is a 1/8" drill bit might be all you need to make mounting holes.
As for using track power, it can be as simple as the following.
The diode converts the AC to DC since LEDs run on DC. The resistors limit the current to the LEDs. The value of the resistor depends on what kind of track power you use (command, conventional). If you use conventional the brightness of each LED will be proportional to track voltage which is probably not a big deal...or with some additional circuitry such as a voltage regulator you can take care of that too. In any event, I'd think all the electrical parts would fit in the cavities underneath one track section if you use, say, MTH Realtrax.
I think the biggest hurdle is to take the mental leap of faith that this can be done! I think it might take a few tries requiring some sacrificial track sections and there are some additional electrical details that need clarification but that's what the OGR forum is about! There are several guys on the Electrical Forum who would also have ideas if you do choose to proceed.
