Blowing on soup cools it
How moving air removes the warmed layer that was insulating the surface.
Blowing on soup cools it because the breath sweeps away a thin layer of air sitting on the surface that has already warmed up and is quietly acting as insulation, replacing it with fresh cooler air that can pull heat away from the soup all over again.
A blanket of warmed air forms in seconds
A bowl of soup left sitting still does not lose heat at a steady rate for very long. The instant it is poured, the air touching its surface begins to warm, taking up some of the soup's heat, and within a second or two that air is no longer very different in temperature from the soup itself. Once that thin cushion of warmed air is sitting on the surface, it behaves much like the boundary layer that clings to any surface a fluid moves past, and it stops being an efficient path for further heat to escape. Heat only moves quickly from hot to cold when there is a real difference in temperature to push it along, and the air right at the surface has largely closed that gap. The soup underneath is still hot, but the layer between it and the room has become an insulating blanket, formed by nothing more than still air warming against a hot surface.
Blowing across the bowl physically removes that layer, pushing the warmed air sideways and away and putting fresh, cooler air against the surface in its place. That restores the full temperature difference that drives heat away quickly, at least until the new air warms up too and the cushion starts building again. It is also why one long, steady breath cools a spoonful of soup better than a string of short puffs. A short puff disturbs the cushion for an instant before it starts reforming, while a sustained stream keeps sweeping cool air across the surface for as long as the breath lasts.
The cushion holds moisture as well as warmth. Hot soup is constantly giving off water vapour, and each gram of water that evaporates carries away about five times the heat it took to bring that same gram from freezing to boiling. Still air over the bowl soon becomes damp, which slows further evaporation, and a breath that sweeps the damp air away speeds up this second kind of cooling at the same time as the first. The steam that bends away from the bowl when it is blown on is a visible trace of both.
Fanning a hot face in a warm room
Waving a hand fan in front of a hot, sweating face on a warm day produces a cooling sensation even though the air it moves was sitting in the same warm room a moment before. The fan disturbs the thin layer of air right against the skin, air that has already picked up warmth and moisture from the body and is no longer drawing off much more. Sweeping that layer away and replacing it with drier air from further off lets the skin lose heat properly again, both directly and by giving evaporating sweat somewhere less saturated to escape into. A face fanned continuously stays cooler than one left still in the same room air, for the same reason a bowl of soup blown on repeatedly cools faster than one left untouched: both are denied the chance to build their own insulating cushion.
Shape plays into it as well. A wide, shallow bowl loses heat faster than a narrow, deep mug holding the same amount of soup, because the bowl offers a much larger surface for moving air to reach, while the mug hides most of its liquid below a small opening where the warmed air settles and stays put. Doubling the diameter of the surface, with the same amount of soup, gives four times the area for that exchange to happen across.
Draughts, stirring and the hot spoonful underneath
The same effect reaches well beyond a bowl of soup, since anything relying on still air to hold its heat depends on this cushion, and anything trying to shed heat quickly is fighting it. A hot drink left uncovered in a draughty room cools far faster than the same drink in a still corner at the same room temperature, because the moving air never lets a warmed cushion settle over the surface. A cook checking whether a sauce has cooled enough to taste often stirs it first for a related reason: stirring mixes the cooled skin of liquid back into the bulk and brings fresh hot liquid up to meet the cooler air.
Blowing mostly cools the surface in the short term. The liquid underneath loses its heat only as fast as that heat can reach the top, so the first spoonful from beneath a blown-on surface can still catch a mouth by surprise after the top has visibly stopped steaming. Still air, given a moment to warm, insulates remarkably well, which is also why the warmest bowl of soup at a table is always the one nobody has touched.