1. The sort of beans used in the $3 cup of coffee are probably quite expensive. E.g., the espresso beans that my local coffee shop uses retail at about $1 per espresso drink ($20/lb of beans with 20 or so double shots in a pound). Plus electricity, amortized cost of the espresso machine, and cost of milk. In the end, a home-made cappuccino of equivalent quality costs about $1.20 - $1.25 to make at home.
2. It takes me about 10 minutes to make a cappuccino and clean up the mess. And I'm pretty experienced with a nice setup. I am paid a lot more than $6/hour. Hell, my college side gig paid a lot more than $6/hour.
3. Coffee shops are a nice place to get work down outside of work (eg side projects or self-learning). Especially relevant for those with kids at home.
So for a $3 drink, he's paying $1-$2 for an hour or more of shared work space and 10 minutes of saved labor time.
That seems like a fair deal.
And this is assuming that it's even possible for him to make the drink he wants when he wants it. If he works in an office without an espresso machine, it might just be straight up impossible for him to make the coffee he wants when he wants it.
Now, maybe Folgers from a drip machine should be good enough. But if we're going to be pragmatic robots, we should really just buy super cheap caffeine powder in bulk and mix it into our free water.... Folgers is for snobby suckers. Plus, I bet you have some hobbies/mini-luxuries I don't care about and could scoff at ;-)