It means the person came from a very poor family who grew up mainly among low income farmers.

modest upbringing = growing in a family that is not too rich, so quite modest / without too much money farming community = they were located in the country/farmers area so this person's family was not very rich and they lived in an area where people were mostly farmers

To add to what Angie and Alice have said I think a good way to distinguish between farming communities and city dwellers is to describe them as rural (farming communities) and urban (city dwellers). However this is not to say that either community is high or low income, rich or poor. If you would like to find out more about me please visit my teacher profile page https://www.italki.com/teacher/4742369 :) And well done! :) I see you are learning two languages!