Spock can’t help but notice how casually McCoy and Captain Kirk touch each other. A slap on the back, the clasping of a shoulder, the casual way McCoy leans against Kirk on the Bridge. The touches seem more than friendship, but less than lovers. Nobody else on the Bridge shares the same propensity for touch.
He voices his observations exactly once. Kirk is amused, McCoy is confused, and now they’re both aware of it, too. Very aware.
Jim figures it out first, and coaxes Bones into realizing (and admitting) to the mutual attraction. Laughing, Jim tells his new lover that they have Spock to thank for their current arrangement. McCoy rolls his eyes and makes a smartass remark about how they should show their appreciation. Jim takes it as a personal challenge, and since McCoy is rather fond of Spock himself, he can’t even be jealous.
So Kirk and McCoy start to gentle Spock, to get him used to touches from them, until it becomes commonplace. And finally, Spock understands why they were touching in the first place.