There are three main pieces of evidence to support the Kinetic Theory of Matter

1) Brownian Motion 2) Diffusion 3) Thermal Expansion

Brownian motion: small particles (dust, pollen, dye, etc.) move randomly when viewed in a microscope

Einstein wrote a paper about this in 1905, explaining that the particles are moving in response to unseen collisions with the water molecules in which the dust is floating. Because the dust moves randomly, we can infer that atoms and molecules do as well. Brownian motion provides evidence that atoms exist and that they move randomly.

(read more about Brownian Motion: link)

(see an animation of Brownian Motion: link)