@Zaku‘s response hit the high points. There’s also one other effect at work here, which is Bernoulli’s Principle. This also applies to @zenvelo‘s response, but in an opposite way.
When you blow on hot food to cool it, you purse your lips and blow ‘hard’ (or as hard as you choose to, anyway) which actually cools the air that you’re blowing from your own body. (It also helps to collect more surrounding air to move with your own internally-generated volume of air and CO₂, and tends to cool that air slightly, too.) So, the harder you blow (and the bigger volume of air that you blow), the more cooling can take place, at least to some practical limits. (If you could blow hard enough to re-freeze the food, then you would probably also blow it all away in the process.)
The way this applies to @zenvelo‘s response is that when you breathe on your hands to warm them, then you don’t ‘blow’ air from pursed lips: you don’t want cooler air! So you exhale open-mouthed (mostly), which tends to keep that exhaled breath at or near your internal body temperature (warmer than your hands, anyway).
I’m sure that evaporation of water vapor (steam) from the hot food is also affected by the blowing of any air over the hot food, as removing that heated vapor-laden air (even by relatively warm air or breath) keeps the temperature differential (Δt) between the food and the ambient air as high as possible, so that evaporation can continue to occur at the fastest rate possible under those conditions. Still, for most foods, most ambient conditions and most mealtime-likely time horizons, it’s the cooler air doing most of the work.