So I put together a quick A-E Listing, so you can match it up with the other side easiest. Short answer, power to A and C should result in it spinning
The A and B are connected by the centrifugal switch, which is used to connect and disconnect the start windings. Once it starts spinning up, the weights get pulled in, and B disconnects, taking the start windings out before they can burn up. (Start windings are the thinner darker coils)
C gives power to the thermal overload, which then looks to branch down on the left side into both the start and run windings. A connects to one end of the run winding. One wire on the bottom of the thermal overload is the other end. (Run windings are the brighter copper thicker coils).
I would clean up the tabs at the end of A and D, and make sure the contact between A and B is clean. I would also make sure the springs on the rotor are bending easily enough for the weight to move back/forth.
You should be able to get continuity between C and one of the bottom wires off the thermal reset. There should be continuity between B and the other wire from the reset. If not, that is the reason it is not working (the overload disconnected inside, and either needs to be reset, or replaced if broken).
Edit: Attaching the picture helps...
