Weather finally cooperates with NASA, ISS resupply launch successful

Gravity. And if that were the only force at play then there would be also be a tiny bit of atmosphere out there too. Not much mind, as the higher you go the thinner the atmosphere gets.

However, unlike the ISS, the atmosphere is made chiefly of air, and as such blows and boils away very easily. (See Solar Winds, etc)

Gravity on or very close to Earth surface is 9.81 m/s2. Gravity on Low Earth orbit (160-2000 km) above Earth surface is 9 m/s2, that's where ISS is. Now would you please tell me what force exactly is keeping ISS on the orbit and why it does not fly away

Centripetal force. The ISS doesn't circle Earth at 7,660m/s just for fun; that 7,660m/s is fun is just a bonus.

Anyway, the ISS averages an altitude from 6,772,000 meters from Earth's core, i.e., it follows a circle of radius 6,772,000 meters. Its average velocity is 7,660m/s.

Centripetal force equals (Velocity squared) / radius, or: (7660m/s ^2) / 6772000m = 8.664m/s/s

Considering the rounding in my math and approximation of velocity and radius values, that's satisfactorily close to your calculation of 9m/s/s for gravity at the ISS's altitude. Gravity is balanced by a satellite's centripetal force. To climb higher or escape, you'd need to move faster so centripetal force exceeded local gravity.

just like air on same altitude apparently did.

The air did not fly away like the ISS. The mean molecular velocity of air at temperatures near Earth's upper atmosphere is around 300m/s, (unsurprisingly) the local velocity of sound. 300m/s is nowhere near orbital velocity, so generally air wants to fall to the ground. Air being compressible, you find about 90% of all air molecules within 20-ish kilometers of the surface and no inclination to fly off into deep space. Generally, getting air to fly off into space requires either:

1) Packaging it in a rocket and launching it to at least 7800m/s, or 11,100m/s if you don't want it to come back

2) Having it circulate to the edge of the atmosphere, suffer improbable collisions with several other air molecules that boost its velocity several standard deviations above the mean, and then get slapped around by solar wind until it exceeds 11,100m/s

It is considerably easier for a puny planet like Mars to lose its atmosphere than a chunky planet like Earth because of the difference in escape velocities.