Obviously, it is caused by the difference in pressure between the inside of the tire and its surrounding environment; but how specifically is the air escaping?
Air escapes via multiple mechanisms.
Diffusion through the material: Most materials have some permeability to air. Air molecules can fit between the rubber molecules. The N2 and O2 molecules are so few and far between that they don't interact. The average force on a molecule is zero as it randomly walks through the rubber, but the concentration difference means that by chance more will leave than enter. Air escapes ~ pressure difference, so double the pressure difference and twice the mass of air will leave per day.
Diffusion through a tiny (nano-sized) pore: Similar mechanism, but the loss rate is higher as there is more freedom for the air to escape.
Diffusion through a small hole (~100nm to a few 100 um): Air molecules now interact since the mean free path is smaller than the hole size. Air acts as a fluid and flows out smoothly. Air escapes ~ pressure difference.
Larger hole (above a few hundred um): Air flows out like a fluid, but this time it is turbulent and makes an audible hissing sound. There is usually no smooth transition to turbulence as you increase the hole size, it happens suddenly the exact transition is somewhat unpredictable. Air escapes ~ sqrt(pressure difference).