Caffeine is a methylxanthine alkaloid, familiar as a stimulant and used topically for quite different reasons.
Vasoconstriction
Caffeine narrows blood vessels. Applied to the under-eye area, this temporarily reduces the volume of blood in the superficial vessels, which is relevant to two things at once: puffiness caused by fluid accumulation, and the blue-violet shadowing that thin skin over congested vessels produces.
This is why caffeine is close to universal in eye products, and it is one of the better-founded reasons an ingredient appears where it does.
The important limitation
The effect is temporary and it is not corrective. Caffeine does not change the structure of the skin, the thickness of the tissue, or the amount of pigment present. It reduces the appearance of a symptom for a period after application.
That makes it genuinely useful and worth being honest about. Dark circles caused by pigmentation rather than vascular congestion will not respond to it at all, which is part of why products aimed at the eye area frequently combine several mechanisms.
Other reported activity
Caffeine has antioxidant properties, and is studied for effects on lipolysis, the breakdown of fat, which is why it appears in body contouring products. The evidence there is considerably weaker than for vasoconstriction.
Absorption
Caffeine penetrates skin reasonably well for a water-soluble molecule. Systemic absorption from topical cosmetic use is minimal and not comparable to drinking coffee.
