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.