That's a modern-transportation perspective. Unless the siege happens to be on a waterway and you supply that way.
Land-based logistics for pre-modern armies was _hard_. The standard solution was to keep the army on the move and forage as you went, so you kept stealing food from new peasants, but that's not an option for an army laying a siege. On the other hand, if the castle/town knew a siege was coming they would move things like livestock from the surrounding countryside inside the walls, and the grain stores were likely _already_ inside the walls.
https://en.wikipedia.org/wiki/Siege_of_Nuremberg explicitly mentions supply as a problem for the besieging army in that case, and that's an early-modern army, not premodern. http://hundredyearswar.com/Books/History/Sieges.htm also says "the siege was often lifted because the besieging commander ran out of food to feed or money to pay his troops".
In practice, what this meant is that you would not even start a siege unless you had some idea that you could ensure supply, which was the common situation.