How does single-phase AC synchronous motor work?

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The wikipedia link in the answer you dismissed to ask this question gives you the answer: "A single-phase (or two-phase derived from single phase) stator winding is possible, but in this case the direction of rotation is not defined and the machine may start in either direction unless prevented from doing so by the starting arrangements "

When AC is applied, the motor is unstable until it starts rotating, so it starts rotating (maybe a butterfly in China flaps its wings). It could (in theory) stay stationary, but only in the manner that if you stand a broomstick on end, it could stay balanced.

Maybe a better example is a piece of paper - stand a piece of paper on edge, and it falls one way or the other. Once it starts falling one way, it keeps falling that way unless you do something like blow or push it back the other way. The motor is unstable, it starts rotating one way or another, and once it starts rotating, it keeps going. You can push it back the other way - if you take one of these and wedge something in the mechanism so it can't continue, it will reverse.

If you want it to only rotate one way you simply build a ratchet or other clever mechanical in the mechanism so if it starts backwards it immediately reverses.

For example, this sort of fishtank circulation pump: http://www.tunze.com/149.html?&user_tunzeprod_pi1%5Bpredid%5D=-infoxunter052

Obviously, a propeller that runs backwards doesn't do what you want, but if the motor starts backwards, what happens is that rather than pushing water out the front, the rotor pulls itself forwards out of its housing. When it does that it then hits a carefully positioned part of the housing in front of the propeller, and bounces off. It's now turning in the opposite direction, and carries on doing so. When it's turning in that (correct) direction, it's pushed back in the housing, clear of the thing that it bounced off, so continues to turn in that direction.