"It gives me energy"
The word most people use for the pick-up is energy. You're flagging and the coffee tops you up, as if you'd been running low on something and it was fuel.
Caffeine has no energy in it. A black coffee contains almost nothing your body can burn. What it does is act on a chemical signal in the brain, and it's worth knowing roughly how, because it explains where the tiredness goes.
Through the day, as you stay awake, a substance called adenosine builds up in the brain, and it's one of the signals that produces the feeling of sleepiness. The longer you've been awake, the more there is, and the sleepier you feel. Sleep clears it. Caffeine works by sitting in the places where adenosine would act, so that the signal is blocked. The adenosine is still there. The sleepiness it would produce isn't felt. That's the pick-up. Nothing has been added. A message has been muffled.
The message doesn't go away while it's muffled. Hans-Peter Landolt and colleagues in Zurich gave nine healthy men two hundred milligrams of caffeine at ten past seven in the morning, about two mugs' worth, and then measured their sleep that night, sixteen hours later. Compared with placebo, they slept less and their sleep was less efficient, and the deep, slow-wave part of sleep was reduced. Caffeine was still measurable in their saliva at eleven at night, at a low level. The authors suggest that either the low level was enough to affect sleep directly, or that having caffeine in the system during the day had slowed the build-up of sleep pressure, so that there was less of it at bedtime. Either way, a morning dose changed the night.
So the pick-up is a loan. The sleepiness you'd have felt at three is deferred, not cancelled, and the deferral is paid for with a lighter, shorter night, which produces a bigger shortfall the next morning, which needs a bigger cup. That isn't a moral point. It's the mechanism, and it's the same for everyone who drinks the stuff regularly.
I'd add my own reading of why the word energy sticks. When you're sleepy, everything feels effortful. Remove the sleepiness and the effort drops, and it feels as if you've been given something. What's happened is that a brake has been taken off. The engine wasn't refuelled.
The honest version of "it gives me energy" is "it stops me feeling how tired I am". That's a useful thing on the odd occasion, before a long drive, say. As a daily arrangement it means you're never sure how tired you actually are, and so you never fix it.