The word "board" in the previous sentence relates to the food that inns were in the business of providing it. Travelers paid extra for their meals, but food was to be had at any place that deemed itself worthy of the name "inn." (Those that wanted only a room could get just that too.)
The "board" in bed and board (or room and board) refers to the board table or sideboard where food was laid out. Common usage came to shift this meaning away from the furniture itself to encompass the food served from it.
More to the point, the term "board" comes from the eating table. Before power tools, it was a great and lengthy effort to make smooth-hewn tables; people would make do with as few pieces as possible. Usually a table was just one board, sometimes two, set on trestles, making a long narrow surface to eat from. Coming to dinner was called "coming to the board," a table cloth was referred to as "board clothes," and when hired help or an apprentice came to stay, they paid in cash or service for their room (where they slept) and their "board" (what they ate).
Note: the American colonials hated making boards suitable for tables so much that they often used split apart shipping crates; there are still examples to be found which have the painted names of the master of the house and the shipping agent/company on the underside.
