What their version does
They've set up the transistor and it's associated resistors (R5, R4, R1 and the potentiometer) to detect when current is injected into the transistor base from the motor via C5. When you inject current into the base, Q2 turns on, and the LED lights up.
Basically, the chain of resistors from the positive power rail to the transistor base try to turn the transistor on, while R4, from the base to ground, keeps it turned off.
When you spin the motor, extra current is injected into the base via the \$470\mu\mathrm F\$ capacitor, until the capacitor charges up. This turns on the transistor, which conducts electricity, and that makes the LED glow.
Why your version didn't work
DC motors work because they have a commutator and brushes. A commutator and set of brushes is basically a switch that's actuated by the motor that always energizes the correct set of motor coils in the correct direction so that the motor turns.
With an inexpensive motor, the commutator doesn't always make contact -- sometimes it does, indeed, open up, and this would be just like your SW1 opening up.
They've chosen what is, in my opinion, an overly complicated way of demonstrating what they're trying to show. For the right motor/LED combination, you should be able to just connect the LED across the motor, spin it fast enough and in the right direction, and see it light up. If you use two LEDs in anti-parallel, then spinning the motor in one direction would light up one LED, and spinning up the motor in the other direction would light up the other LED.
It could be that with the motor they've chosen you'd have to be especially good at spinning the motor to get it going fast enough -- to do that really "right" you'd need a gear train and a crank, and then you'd pretty much have a dedicated generator, or at least a project that's more mechanical than electrical.
But, if you connect the circuit as they've shown, it should work.
If you want a deeper dive, here's a video I did on how a brushed DC motor works. I can't remember if I went into any depths as to how it works as a generator, but the short story is if you use electro-magnetism to move something, then that same mechanism can almost always be a generator, too.