Why do academics drink so much coffee?
Academics sure do drink a lot of coffee---but do they drink any more coffee than other professionals?
According to Wikipedia, 150 million Americans adults drink coffee daily (out of about 240 million total), a rate of more than 60%. So if most adults drink coffee in America, it's entirely unsurprising that academics are like most other adults and that departments might happily supply this cheap perk---just like many, many other offices do. Similarly high rates of consumption appear to pertain in most other developed Western countries, which are also where the majority of high-impact scientific research is still conducted.
In short: many academics appear to drink a lot of coffee simply because they are typical adults in societies where most people drink a lot of coffee.
There are at least two possible explanations and neither involves coffee. The first is that when you try to force your brain to work it sometimes rebels and leaves you stuck. Taking a break, any kind of break can undo the knot. There are stories (and I've experienced it myself) of going to bed with a sticky problem and waking up with the answer. Your brain, generally, isn't idle. It can seek pathways more or less unconsciously.
The second reason is that in large departments the coffee is in a common room, usually with a table that a few folks can sit around and an adjacent white/blackboard. So you get a cuppa and you sit down and chat about your work. Someone else says something that strikes a chord and you have the germ of the answer. Actually, sometimes when you chat about someone else's work, the key comes to you almost unbidden.
Those who work in isolation don't get the benefit of the second solution, and I found that to be true myself. The colleagues you chat with don't even need to be deeply involved in your own research, but sometimes a hint from "out in left field" gives you the path to integrating the concepts you've been working on.
On the other hand. Coffee is good. Long unbroken hours of intense work are usually less good. Less tasty anyway.
Note that a coffee pot in your own office gives you neither of the above benefits.
There are two components to your question: the actual presence of coffee, and the social aspect of talking about it.
The Social/Humorous aspect:
Coffee, and the need for it, is a memetic joke in Western culture, where the obsession and need for it is humorously exaggerated. This happens in fiction but also is a common trope in the working world, especially in desk jobs, which academia primarily is. Jokes like "Don't Talk To Me Until I've Had My Coffee" are resonant enough to have a McDonalds commercial, and appear on literally millions of mugs.
The actual usage aspect:
Coffee is the primary, non-stigmatized stimulant in wide use in Western culture. It's not even close. 90% of adults in the US consume coffee every day. I think you'd find that it is prevalent in every setting, not just academia. As a software developer, I hear this humor all the time.
Why is it so popular? Well, in a culture that celebrates visible displays of overwork, and in a profession in which both overworking and bragging about it are common (see this question, it's no surprise that it both is widely consumed (to compensate for long hours) and also widely talked about (since consuming lots of stimulants indicates that you're working really hard, of course).
tl;r: toxic culture and overwork.