The diodes are there so that there can be two separate mechanisms for enabling the main headlight relay without interfering with each other. From the labels on the right, it looks like the two mechanisms are the normal headlight switch, and the momentary passing switch, which makes sense. You want each to be able to turn on the headlights independently.
You now say you are looking for more of a low level electrical explanation of what the diodes do instead of the high level conceptual explanation you originally asked for.
The diodes allow either switch to turn on the relay, but still remain independent. Presumably at least one of the switch lines is used elsewhere in a way that it shouldn't be signalled when the other switch is activated. Without the diodes, there would be only a single switch line, and that would be activated identically by either switch, making it impossible for other parts of the system to determine which user action caused the headlights to turn on.
For example, the general headlight switch may also turn on some running lights or maybe a indicator on the dash that the passing switch is not supposed to turn on. With the diodes in there, the general headlight switch line remains unasserted when the passing switch is activated. Each switch can activate the relay, but there is no connected between the switches.
A GPIO high will produce a drain voltage of ~0 at the first FET. Without the divider, this produces essentially VDD between gate and source of the second FET. A VDD of 30 volts greatly exceeds the maximum gate voltage, typically 20 volts. With a 2:1 divider, a VDD of 30 volts produces a gate voltage of 20 volts, while a 6 volt VDD will produce about 4 volts.
One purpose for that diode is to block a leakage current path if the +3VRUN rail is powered up but that the +5VRUN rail is not powered up. Allowing an unpowered rail to get leakage current from other powered circuits can lead to operational problems and potentially result in severe latch-up in circuits referenced to that rail when they become powered.