Hot tools/parts are cooled by removing heat. This can be done by transfer through absorption of heat to a cooler material as is done with flood coolant or by evaporative cooling as is done with mist coolant.
The ability of a material to remove heat by absorption depends upon the specific heat of the material (calories/ºC/gram) x the mass of the material. Flood cooling works because the coolant has a fairly high specific heat and we flood the work area with fresh coolant. It also has a high thermal conductivity which facilitates heat transfer.
Mist coolant has a fairly high specific heat but there isn't much mass. However, the mist coming in contact with the tooling and/or work evaporates. While the mist has a specific heat of around 1 cal/ºC/gram, it has a heat of vaporization of around 550 cal/gram and is therefore capable of efficient cooling. The compressed air in the stream aids in chip removal.
Using oil serves primarily as a lubricant. Its specific heat is much lower than water based coolants and its thermal conductivity is lower as well. Its lubricity does assist, though,by reducing friction and resultant heat generation.
Using compressed air does not provide much in the way cooling by virtue of its low specific heat and low mass but it does serve another important function which is chip removal. We use either water or oil to quench steel to harden it because of their ability to quickly pull heat away from the steel. Subjecting the steel to a blast of cold air simply will not do the job.
