Question

A sprayer works on the principle

Answer 1

The Swiss physicist Daniel Bernoulli formulated the Bernoulli's Principle. It describes the way in which fluids behave. According to it, pressure decreases as the speed of flow increases within a stream of fluid. Similarly, pressure increases as the speed of flow decreases.

A sprayer used for perfumes or insecticides works on this principle. The air flows at a high velocity through the tube when the plunger is pushed in. A region of low pressure is created just above the tube due to the flow of air at a high speed. Due to this the perfume or insecticide leaves the tip of the tube through the nozzle as a fine spray which is carried by fast moving air.

Answer 2

The  answer of your question is "Bernoulli equation" which talk about fluids behaviour . It stated that pressure and speed are inversely  proportional to each other. So if the pressure increase the as the result of decrease in flowing speed of fluid and vice versa.
In case if spray air flows through the tubes in the high speed result in decrease in pressure due to which fluid through the tubes passes in the form of fine spray and then those spray travel fast due to high speed of air.