Trying to understand PWM / solve relay buzz

Hi Everyone. Im working on a 3position switch hookup for some aux driving lights and turns out my 'hi beam on' signal wire is PWM. I understand why that makes the relay buzz but im not sure how to address it. I have found diagrams like this which shows a capacitor but I don't understand why the capacitor goes to ground vs being 'in line' to the circuit. So that's my first question, why does this work? (image grabbed from a post by BobK) From other 'anti flicker' setups I have seen I do know this works but that's not good enough, I want the why too


My next question is how would I wire this up with a 3 position switch (maybe knowing the why would help) In position 1 the input is 12v off the battery, position 2 is off, position 3 is PWM.
If I wired it up 'as show' would there be a problem when I had the switch in position 1?
Thanks guys! And sorry if this is a dumb question, I tried googleing around but I lack the vocabulary to ask the right question.

 

Thanbs Bob, I understand the capacitor holds the charge I guess maybe I don't understand how it works. I THOUGHT Volts come in one end and go out the other so it would be providing a charge on the ground side in the diagram. What I think im reading from you is it provides the charge on the 'in' side and the ground side is there for …. overflow?
If that's the case then this works just fine on the PWM and steady 12v circuits?
Is there anything you can point me at to understand sizing the capacitor? Other posts for headlight antiflicker says to use 4700uf 35v capacitors. Don't know why that specific size and im assuming I don't need that big of one as im driving a relay vs a whole headlight.
Looking at the diagram again its from a post from 2016 https://www.electronicspoint.com/forums/threads/effectively-defeating-pwm-on-jeep-wrangler.279513/
THANKS

 

Thanks Bob for straightening me out on the capacitor and the math! Makes perfect sense.
tim

 

Thanks Duke, I don't know FET's (though I did fine what thats short for) so ill have to go look in to that some.
For the 'drop the capacitor' idea, my goal was to make the buzz go away, first I can hear it though maybe I can wrap it in something and deaden it but second, I didn't think that was good for the relay and would meaningfully shorten the life of the relay.
thanks
tim