formal. : to become ill Dozens of workers fell sick from exposure to the fumes.

Is fell sick correct?

In the US we say "got sick" and "fell ill". Although, "falling ill" implies that it's just the start of something that's going to turn out to be much more serious and long term - Like: "He got sick with the flu, last week," as opposed to "She fell ill with cancer, as soon as they returned from the trip."

How did you fall sick?

Whenever the normal working of our body system gets disturbed, we feel sick. It can happen when a bacterium, virus, etc, enters our body or because of unhealthy living practices like lack of exercise or intake of drugs/excessive sugar/salt.