Short answers, pulled from the story.
John Tyndall observed faint blue-tinted light scattering off nanoscopic particulates in purified air in 1869. He conjectured that a similar process gave the sky its blue hue but could not explain why blue light was preferred over red.
Lord Rayleigh published two papers on the color and polarization of skylight in 1871 to quantify Tyndall's effect. These papers described tiny particulate volumes and refractive indices within water droplets.
Blue light wavelengths scatter more strongly than longer red wavelengths due to the inverse fourth power dependence. Shorter blue wavelengths are scattered out of the direct path of sunlight reaching the ground, causing the diffuse sky seen in daytime to appear blue because of this strong wavelength dependence.
Rayleigh scattering causes acoustic wave damping and phonon damping in glasses at low or not too high temperatures. Higher temperatures obscure the Rayleigh-type regime with anharmonic damping that becomes increasingly important as heat rises.
James Clerk Maxwell proved the electromagnetic nature of light in 1865 which allowed Rayleigh to show his equations followed from electromagnetism by 1881. In 1899 he demonstrated that these principles applied to individual molecules establishing the basic scientific model for the color of the sky.