Can someone explain how exactly a Spy "helps twart other spies"? Does he do that by his mere existence, or by being in proximity of the enemy spies, or how else? Is the effect quantifiable, and does it multiple with the number of one's own spies?

I don't know if spies help thwart enemy spies, I think the buildings (especially the last anti-spy building) helps mostly.
And having a higher spy-point ratio also helps.

But if domestic spies work as counter-intelligence, I'd love to know about it as well :)

I think spies do help thwart enemy spies. I think it's wise to leave spies in your most important cities to help prevent enemy infiltration. You'd be surprised how often you catch them if you have spies around.

I think spies do help thwart enemy spies. I think it's wise to leave spies in your most important cities to help prevent enemy infiltration. You'd be surprised how often you catch them if you have spies around.Well, I've played a few games without being bothered to buld spies and plenty of enemy spies get caught.
That's why I want to know if it helps or if it looks like they help.

It would be silly if we were building spies for our cities and then found out that they dont have any effect.

It would be nice if someone who designed the game or can read the code could tell us.

That a spy "helps thwart rival spies" is a direct quote from the Civilopedia:

Special Abilities:
- Invisible to All Units
- Can Explore Rival Territory
- Helps Thwart Rivals Spies
- Starts with Commando