It does seem strange that we have to blow on things not only to make them warmer but to make them cooler. There is, of course, a very good reason for doing this.
When we blow on our cold hands we are using the breath from our body and this is much warmer than the outside air.
Therefore we feel that warmth when we blow.
When we blow to cool ‘things, though, the position is reversed. The heat from tea or coffee, for example, is much warmer than our breath. So to make sure we can drink a cup of hot tea or coffee without burning the mouth, we blow on it.
As our breath is much colder than the heat from the hot drink, we are really using cooler air to blow away that heat.