It means to hang out a sign advertising, their business. Basically it means to establish their business. It's a slightly old fashioned way of saying things.

I think ‘hanging out a shingle’ is more common than ‘hanging a shingle’. The term is American and In rural America before there was mobile communication it was often used to identify a business operating out of residence. This included professional businesses such as lawyer, doctor, veterinarian, dentist and accountant, but also things like well-digging, tree-felling, and snow plowing. Now people don’t want customers showing up at their house unannounced and the phrase refers to professionals opening up solo practices.

A "shingle" can be a wooden board, about 30 cm by 100 cm, used for covering the sides or roof of a wooden house. To "hang out a shingle" means to advertise a business by putting out a sign. Often it means a small business, operating out of somebody's home. The sign is a small board with the name of the business, hanging from a post. "Family doctors tend to hang their shingles in rural and poor areas" combines two ideas. A "family doctor" means a primary care physician--years ago. a "general practitioner" who did everything: prescribe medicines, sew up cuts, set a broken bone. "Hang their shingles" suggests a single doctor, possibly operating out of part of their home. The sentence suggests that you will still find this kind of doctor in the US, but you will mostly find them in rural and poor areas. In a big city, doctors are apt to belong to group practices or big clinics, working out of a big medical office building.