A fin makes something cool faster
Trading area against temperature, and why it works.
A fin makes something cool faster by giving heat far more surface area to escape through than the object's original outer shape ever offered, and because heat loss to the surrounding air scales with exposed area, more area moves more heat away for the same temperature difference.
McGee's On Food and Cooking turned out, once I actually sat down and read it properly, to explain heat transfer through a joint of meat with more clarity than any of the textbooks that were supposed to be teaching me the subject.
Heat can only leave through a surface
Heat crosses from a warm object into the air only where the two touch, so the rate it leaves depends heavily on how much surface the object presents. A compact, blocky shape has little surface for its size. Adding thin extensions changes that quickly: a flat plate 100 mm square has 10,000 square millimetres of face, and ten fins 30 mm tall standing across it add 60,000 more, counting both sides of each, so the same footprint now has seven times the area in contact with air.
Heat, following the material's thermal conductivity, runs along each thin fin from base to tip, leaking sideways into the air brushing past every face as it goes. That is why fins are cut from a good conductor such as aluminium or copper. A poorly conducting fin stays hot near the base and cool toward the tip, and the cool part of it adds area that has too little temperature difference left to drive heat into the air.
A fin changes how quickly heat leaves an engine or a processor, and leaves the amount being generated exactly where it was. It also only helps while air can reach it, so a fin coated in dust or grease loses most of its advantage.
Washing on a rack and washing in a pile
Wet washing spread across the rungs of a rack dries within an hour or two on a breezy day, because air reaches every damp surface directly. Bundle the same clothes into a tight pile and the outer layer dries while the middle stays damp for most of the day, since moisture from inside has to work its way out through layers of already-damp fabric.
A fin spreads a warm surface out the way a rack spreads laundry, and the physics of evaporative drying and convective cooling are close cousins, both governed by how much surface the air can reach. Move the rack indoors into still air and drying slows even though the area is unchanged, because the air has to be moving across that area too, a point that matters as soon as fans enter the picture.
Orientation matters for the same reason. Vertical fins let warmed air rise freely along each channel, drawing fresh air in from below, while the same fins turned horizontal hold warm air between them and give back part of what the extra area gained.
Thin fins, sensibly spaced
Because the benefit comes from exposed area, a good fin design keeps each fin thin enough that heat still conducts quickly to the tip. A thicker fin moves heat no faster and wastes material. Fins spaced too tightly run into a different limit, where they start interfering with each other's airflow, the point at which a drying rack crammed with too many items would also start drying slower.