Ambassadors (and all other diplomats, even messengers too) used to represent Kings in foreign countries, and also used to take decisions in some cases (like entering into a treaty) on behalf of their kings. This is historical basis/foundation of diplomatic immunity. Now they represent sovereign states.
Functional foundation is that countries soon found out that diplomatic immunity and privileges are crucial for diplomats in discharging their duties. Without these privileges everybody will be at loss (both sender and receiver states) because such privileges are meant to be reciprocal.

I hope I have explained it sufficiently in few short sentences. You can also search historical events in relation to these foundations i.e. how these come about etc.