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I've been watching videos of various dignitaries (heads of state, heads of government, important officials, etc.) arriving for state visits in foreign countries. I noticed that upon disembarking from their car/ship/aircraft, the visitor and their host would inspect a guard of honour together.

It appears to me that inspecting ceremonial guards is now a global practice for visiting dignitaries. So the question is: what are the origins of this practice? Where and when did this practice become common? What was the original purpose of showing off soldiers in pretty uniform?

When a foreign dignitary was on a State visit the host was responsible for the security of the visitor. This was guaranteed by providing a military formation to do so and this was usually really inspected by the visitor.

Today Americans – for example – do not trust this old tradition and provide their own security (largely). But it is still a part of the traditions of diplomatic protocol and seen as a big insult if such a defilee is not offered. Each country is allowed to have its own tradition for that, but you have to offer something along those lines.

Then there is some additional symbolism meant to be conveyed:

The symbolic intentions of a reception with military honors are above all:
Representation of state sovereignty to the guest (demonstrating state freedom and ability to act),
Presentation of the weapons during the front line walk as a mixture of demonstration (I am armed) and presentation of the weapon for checking for ammunition (I have not loaded).
Achieving a good impression with the guest of the discipline and the level of training of the own armed forces (protocol units are therefore usually subjected to special drill and hard drill training).

Wikipedia