It is only a light phenomenon....
The light that we receives on earth
comes from the sun. The white light
produced by the sun is a set of all
colors whose sum forms the white.
In other words the sun produces the
red, the green, the yellow, the blue....
and all these mixed colors
form the white. Every color
has a wave length well defines.
For example, The blue has a wave
length of 446 nm whereas the red
has a length well of 750nm

When the sun light enters in the atmosphere, it is going to go back in contact with the entities present in the atmosphere as air molecules (oxygen, nitrogen, carbon dioxide, ozone....), the drop water, and the dust. All these species are therefore obstacles to the passage of light.

Some colors of light pass these obstacles better than other. The air molecules have the good dimension to distribute the shortest wave lengths of light, the purple, indigo and blue in this case; the longer wave length, as the reds, are not distributed more or less by these air molecules. 

It is therefore a mixture of purple, indigo, blue, green and a small fraction of the other colors that being distributed in the whole sky, conferring him this blue color that you know well.
Without this atmosphere the sky would be black and you could see the stars in full day.

Yes but the sky is more or less blue according to the days....

The air molecules are responsible of the blue color of the sky. On the other hand, the water molecules and the particles of dusts amplify the diffusion of the green and the yellow, giving a clearer hue to the blue. The absence of dust and drops of water allows the reinforcement of the blue radiation.