The sky looks blue because air molecules scatter the blue part of sunlight in every direction far more strongly than they scatter the other colors — an effect called Rayleigh scattering.
Sunlight looks white, but it is really a blend of colors with different wavelengths. When that light enters the atmosphere, it runs into gas molecules that are far smaller than the wavelengths themselves. Particles that small do not scatter every color equally — they scatter short wavelengths much more. Blue gets thrown in all directions, again and again, until it is coming at you from every part of the sky.
That is the trick of it: the air is not blue, and nothing is painted. Your eyes are picking up blue light that has been bounced all over the sky before it reaches you.
Near sunset, sunlight has to cross much more air to reach you. Along that longer trip, most of the blue scatters out of the beam before it arrives. The longer wavelengths — reds, oranges, and yellows — survive the trip better, so those are the colors that dominate the evening sky. Same physics, longer path, different result.
See the short version in the LOE PRESSURE "Things You Didn't Know" series — Episode 22, why the sky is blue.
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Sources: NASA, Rayleigh scattering and sky color explainers; NOAA SciJinks.