They don't usually form out of nowhere that quickly, but they grow upwards that quickly in case of unstable air. Unstable in this context is a function of the temperature gradient of the air mass with altitude, a bit simplified: If the temperature in an air mass is decreasing rapidly with altitude you have unstable air.

What happens is that some parcel of air is pushed upwards by something (e.g. surface heating, wind over surface features, a cold front moving underneath). That parcel of air will cool down due to decreasing air pressure. But if that cooling down happens at a slower rate than how much cooler the surrounding air is, it will still be warmer and keep accelerate upwards more. This can repeat itself to cause rapidly rising air.

If there is a lot of moisture in that air, the effect is worse. Because at some point that air will have cooled to the condensation level where relative humidity becomes 100%, then the moisture that was in the air will condensate and form clouds. That condensation process will release latent heat, helping the rising air parcel to keep rising further and faster.

And these are exactly that kind of conditions that cause big cumulonimbus clouds to form, which are dangerous even to airliners. Around the tropics, these clouds grow much higher than in more northern or southern latitudes.

Source: The theory class for my airline transport pilot license. A meteorologist would probably tell you this info in much more scientific terms :-)

I've never seen one form out of clear air that quickly, as the pilot said local conditions were. I have flown a high performance jet alongside a rising cumulus and "raced" a column forming on the side that was rising quickly - it won, as in I ran out of climb capability before it did.

It reads like the clouds moved that quickly which would indicate some kind of crazy updraft.

You can see air moving if opaque particles (eg. a cloud) are in the moving stream of air. Otherwise you don’t see anything.

Not in this case, but they can.

Plane condensation trails are a form of clouds.