My father was just told that he has asbestosis. How did he get it?

Asbestosis is not inherited and does not occur on its own. Your father most likely got it from the environment. 

The majority of people in the USA acquire asbestosis from work related exposure. Asbestosis is a chronic condition of the lung which occurs when tiny particles of asbestos fibers are inhaled. Over time, the lungs lose elasticity and breathing becomes very difficult. 

Asbestos in the home for insulation has practically been eliminated in North America. So my guess is that your father probably developed asbestosis from work.  If you have no asbestos insulation at home, find out what work your father did and speak to a lawyer.  He most likely was not given adequate protection and thus inhaled the asbestos particles while working.