IF the switches are simple series SPNO switches as shown and
IF the LEDs are standard LEDs then by definition it is not possible - he is using video fakery and the demo is less than valueless.
"Not possible" is able to be said with certainty because examining even SW1 + SW2 + SW3 in series current can flow only when all 3 are operated.
HOWEVER if the "switches" and "LEDs" are modified then it is possible and it becomesa conjuring trick with rigged props. Still pretty pointless, but possible.
One of various ways:
Each switch contains a current determining circuit.
Each LED contains a window comparator and current bypass circuitry
LED1 lights for I1 or (I1 + I2) or (I1 + I3) or (I1 + I2 + I3)
eg say I1 = 10 mA, I2 = 20 mA, I3 = 40 mA.
LED 1 will light at 10, 30, 50 and 70 MA LED2 will light for 20, 30, 60, 70 mA LED 3 will light for 40, 50, 60, 70 mA
So
mA LEDS lit
10 001
20 010
30 011
40 100
50 101
60 110
70 111
Hopefully that table looks familiar.
There are many other trick ways and it is more likely that the system is "a trick". .
UPDATE:
Professors cam be liars and magicians also, apparently :-).
Again - AS LABELED this breaks the laws of Physics, it's a trick, something special is being done.
{Here is a very long HackADay discussion on this exac topic](http://hackaday.com/2011/12/19/ask-hackaday-troll-physics-edition/?utm_source=feedburner&utm_medium=feed&utm_campaign=Feed%3A+hackaday%2FLgoM+%28Hack+a+Day%29Here is a very long HackADay discussion on this exact topic) by many people. Nobody there is certain how it is done, but some have more ideas than others.
If an excessively enthused person will pay me $1000 and I'll deliver you an essentially identical or better demonstration by end of this year. Pay me nothing and I just may do so anyway :-).
But, be certain, "It's a "magic" trick" [tm] using electronics as its medium. As long as you are aware of that aspect the actual how is unimportant.
Found today in passing.
Inline LED driver
While this by itself does not fi=ully answer the problem it shows what can be done: