The answer
Air molecules scatter blue light much more than red light, so blue comes at you from all over the sky.
Sunlight is a mix of all the colours of the rainbow. Each colour is a wave with its own length: red waves are long, blue and violet waves are short.
The air is mostly nitrogen and oxygen molecules, which are far smaller than a wave of light. When light hits something that small, it gets scattered in all directions, and short waves get scattered much more than long ones. This is called Rayleigh scattering. Blue light at 450 nanometres is scattered about four times as much as red light at 650 nanometres.
So as sunlight crosses the air, a lot of its blue is bounced sideways, again and again. Wherever you look in the sky, some of that bounced blue is heading your way. That is the blue sky.
At sunset the light has to travel through much more air to reach you. By the time it arrives, most of the blue has been scattered away along the path, so what is left is orange and red.
Go deeper
Why? Why isn't the sky violet, if violet scatters even more?
Violet does scatter more than blue. But the Sun gives off less violet than blue, some of it is absorbed high in the atmosphere, and our eyes are much less sensitive to violet. Add it all up and our eyes see the sky as blue.
Why? Why are clouds white, not blue?
Cloud droplets are much bigger than air molecules. Big particles scatter all colours about equally, so the light they send you is still white.
Why? What colour is the sky on Mars?
Mars has a thin atmosphere full of fine reddish dust, so its daytime sky looks butterscotch. Its sunsets look bluish, because the dust scatters blue light forward, toward the Sun's direction.
Keep wondering
Sources
- Why Is the Sky Blue?, NASA Space Place
- Why Is the Sky Blue?, NOAA NESDIS