How does a multi-tap motor speed control work?
Comparing those tables: Note that the speed switch in the circuit you show isn't using L.

A: L+2+3 
B: L+1+3 
C: L+1      (Maybe this is L+1+2 ???)
D: L+1+2+3

0: No connection (or no connection to anything but L)
1: 2+1 (possibly plus a connection to L)
2: 1+2+3 (possibly plus a connection to L)
3: 2+3 (possibly plus a connection to L)


Making them correspond with each other...

C is obviously equivalent to 0.
D is obviously equivalent to 2.


That leaves us with

A: L+2+3 
B: L+1+3 

1: 2+1 (possibly plus a connection to L)
3: 2+3 (possibly plus a connection to L)


We can make those match if we relabel the connections. If we just swap the labels on your terminals 3 and 2, then 

A is equivalent to 3
B is equivalent to 1


If we renumbered them all (your 2 is their 3, your 3 is their 1, your 1 is their 2), then 

A is equivalent to 1
B is equivalent to 3


Pick whichever you prefer; one will switch off-high-medium-low-off, and the other will switch off-low-medium-high-off.

As far as theory goes: I'm not sure either, but let's see what I can do with it.

3 (2->3) appears to be "slow" because power flows through the right half of the bottom coil, and then through the side coil, in series. More resistance, less current flow, less power.

1 (2->1) appears to be "fast" because the left side of the bottom coil, and the side coil, are powered in parallel. Both get the full house-current voltage applied across them rather than the reduced amount of power they got in series.

2 (2->1 and 3) is the tricky one. I am far from certain, so DON'T take my word for it. But I think what's happening here is that, since the middle and right sides of the bottom coil (1 and 3) are now connected to each other,  that loop has a current induced in it by the motor's moving magnets, which creates a countering magnetic field, which acts as a magnetic brake to slow the motor... so fast with a bit of braking equals medium. Seems like an odd solution, but if I'm remembering my freshman Physics at all correctly it might actually be a reasonably efficient solution.

You might want to run this by the physics discussion, to get someone with more recent memory of electrodynamics to check and/or correct that last paragraph.

Gopher baroque...