How does sun heat the air?

Sigh ... there's a great deal of misinformation here. A correct answer is long and complicated, but the short of it is that there are numerous gas absorption bands in the "solar shortwave" -- these range from the Hartley-Huggins bands of Ozone in the UV (responsible for the stratosphere), the Chappuis band of O3 in the mid visible, a variety of weak absorption bands of Oxygen, and then a substantial H2O absorption band at 940 nm ... with increasing numbers of absorption bands in the near infrared due to a variety of trace gases starting with H2O and CO2.

So sunlight can and does heat the atmosphere through direct absorption; absorbing aerosols also play a role.

Nonetheless, as is everybody's direct experience -- a lot of light gets down to the ground on a cloud-free day.