@Angel M has commented well on this issue.
Succinctly, "kindly" within an instruction is a nod to politeness, intended to take the edge off something which is clearly a command, not really a request, not really up for negotiation by the speaker.
It underlines that the speaker is well aware of what constitutes good manners and nevertheless the terms of the following instruction will not be negotiated.
"Kindly sit down, sir" may sound a bit Dickensian as a command but if a person is well-read in English literature they will be aware of this connotation.
"Kindly address your enquiry through the proper channels" - shuts down negotiation while pre-empting any accusation of disrespect.
"Kindly return my lawnmower" would be pretty aggressive in a note to a neighbour.
As a result, inserting "kindly" into a request gives it the construction of a command.
"Kindly keep your opinions to yourself" is an example of how it would be used to intensify a statement from brusque instruction to a threatening command tone.
Therefore non-native speakers or those who have limited
experience of this idiomatic use are risking creating the opposite effect they intend.
Yes, it's true that language evolves and there will be posters who are unaware of this usage who will contradict my post, claiming they've never heard of this so I'm wrong. To them I say, kindly do your homework ;)
The reason I searched out this post is that I am getting tired of these little unintentional command impacts I receive from learner friends and colleagues who use this, thinking they are appearing nice and in fact are appearing to be presumptuously ordering others around. You need to avoid using this phrase with anyone you might guess is widely read in English.
Just avoid it, it's not kindly at all.
The word kindly is useful as an adjective of a person's character, meaning "of a kind disposition". Otherwise as an adverb meaning "in a kind manner". It's a good word in general, I wish we had cause to use it more.
However it absolutely gets my goat when used to soften the tone of a request; kindly stop using it in this way.