Jean Paul Marat is a fascinating figure for many reasons but I'll stick to just your question.

While the majority of the east Parisian mob would not have been perfectly literate, everyone likely would have known someone who was and particularly during this time period there were a great many people who would have been very eager to stir up the mob in exchange for their support too!

This means that while many people were unable to read the writings of Marat for themselves, most everyone would have eagerly listened as someone read them. It was not uncommon during this time in France, and indeed much of the rest of the world, to have daily or weekly readings of various newspapers. Their choice of newspaper would often be influenced by things such as royal decree (sometimes), popular demand(more likely), and personal beliefs. This meant that it would only take a few literate people in the neighborhood(or from elsewhere and interested in the power of public support) to have everyone informed about his writings!

Per sourcing I'm afraid I have little that's used friendly about this topic as much of my knowledge in this regard is an amalgam of passing remarks made in other primary sources at the time but I can try and track down one of them as an example....