The round part that is hanging down in the center is a sequencer, it is what they are using for the time delay. The coil on that should be powered by the generator. The switch side of that sequencer will also have power from the generator, then goes to the coil side of the transfer relay. So when you start the generator it powers the coil side of the sequencer, it heats up and closes the switch side, which sends power to the transfer relay. That sequencer is the same part used in many electric house furnaces. They go bad all the time. The coil will have a hot and a neutral wire and the switch side will have a hot coming from the generator, going to the coil on the transfer relay. If the part is bad, make sure you replace it with the exact same kind= amp rating. Some with have 12 v, 24 v or 125vac coil.
It does look like it has a start cap on the right side, don't know why they would need that. You aren't trying to trouble shoot this while being plugged in are you? that may be pretty confusing.
Check the sequencer operation first. You should have no power to it until you start the generator. Then you should have 120 vac measured at the coil. Each side of the switch should have 120 vac measured to ground . If you don't the sequencer is bad.