Here is my prof's website which I'm pretty sure explains why:
**broken link removed**
THe gist of it, is that the capacitors add a phase shift between current and voltage (in the auxillary winding of the motor) which is required for the motor to run. THis phaes shift is there in a 3-phase source but does not exist in a single phaes source since there is just a single phase. But the capacitance needed for efficient steady state running operation is different than the one needed for best start-up torque performance. So two capacitors can be used, essentially switching the capacitances in and out depending on whether the motor is starting up or running. After that, you can make all sorts of compromises to reduce cost (like just one capacitor for either best startup, best running, or somewhere in between) depending on the application requirement, or go all-out and have both with switches to enable and disable them.
Or apparently from the webpage you can remove them all together but then you have worse performance compared to the other two and there are limitations. I think you need a motor designed for it, since it says something about high resistance in the auxillary windings.
On how the motor itself actually works:
**broken link removed**
THe gist of this, is that if you look at that graph, notice that at zero speed you have zero torque, therefore the motor can never startup on it's own unless you have an external force applied or do something else to it (like the capacitors). All that double rotating field stuff is a calculation method for the motors (it bears similiarities to the way the 3-phase calculations or done see).
I would not say the capacitors don't shift the phase to mimic 3-phase power. I'd say that it would mimic it enough for an induction motor to run. It will not replicate a 3-phase supply from a single-phase supply (you couldn't run other types of 3-phase motors or loads from it.