The diodes let the current in one direction but not the other. On the motorbike the ignition is allowed to run if the gearbox is in neutral or the side stand is up. That could be done by putting the two switches in parallel. However that would make the neutral light come on when the side stand is up. The diode lets the current from the CDI go to either switch, so either switch will let the engine run. When the side stand is up and the gearbox is not in neutral the diode stops the current from the neutral light going through the side stand switch.
When the neutral switch is on, current will flow along the orange route, allowing the starter and the ignition to operate. The current flows through the diode, the upper one in the diagram. (The neutral light will illuminate, but that doesn't need the diode)
When the stand switch is on, the current cannot flow along the blue path, because that would be going through the diode the wrong way. The neutral light will not illuminate because the diode will block the current. (Starting and ignition will be allowed, but that doesn't need the diode)
Similarly, starting is allowed if the stand is up and the clutch is disengaged. Current from the starter safety relay (5) flows though switch 8 (clutch) and switch 9 (stand) and allows the starting. The other diode, the lower one in the diagram, will stop the current from the neutral light from also going through those two switches, so the neutral light stays off.
You can test the diodes as follows:-
Turn on ignition. Select neutral. Leave stand down. Do not pull the clutch lever. Try to start engine. If it cranks and runs, then both diodes conduct in the forward direction, which is correct.
Stop engine. Turn ignition on. Select a gear. Lift stand and pull the clutch lever. If the neutral light light stays off, then both diodes are blocking in the reverse direction which is correct.
As a separate test, if you turn on the ignition but don't start the engine, select neutral and leave the side stand down, the voltage across the stand switch should be about 0.6 - 0.7 V.