It does. Heat conduction is faster when the temperature difference is large. Air soaks up a lot of heat, so still air is a poor heat conductor. If you're blowing it around, you're increasing the amount of fresh, colder air that can interact with the food.
One spoonful and a couple of breaths is small enough stuff to have a relatively small effect and a lot of error margin, though.
Besides evaporation, blowing on something which is very hot will bring it closer to room (or breath) temperature by removing the air right next to it which has been heated up already and is 'protecting' the hot item a bit from direct exposure to the room air. When you blow that hot air away the hot item then touches the room air and heats that up, which cools it down.
The same applies to cold things, you can melt ice cream for example much faster by blowing on it even if you use a fan instead of hot breath.
The simple answer is that it takes time. You've removed it from the hot pile of food and given it a few moments to cool, surrounded by cooler air.
I bet if you did an experiment where you blow on on spoonful while it waits and with the next one you just pause without the blowing, you'd find little difference between the two experiences.
You know how you wind can make you feel pretty cold even when it's warm outside? The effect is even stronger for food, since the ratio of surface area to volume is bigger, and the temperature difference is much bigger.