As part of Ichabod Crane's compensation for teaching the children in the Sleepy Hollow area, he gets his room and board, on a rotating basis, from the farmers whose children he is teaching. Irving notes that "With these [that is, the farmers] he lived alternately a week at a time, this going the rounds of that neighborhood, with all his worldly effects tied up in a cotton handkerchief." One of the reasons Crane has to alternate lodgings on a weekly basis is that his appetite is so huge that he is undoubtedly a burden on his host.
There are two reasons Irving includes the detail of Crane's belongings being carried in a handkerchief. As an itinerant (someone who has to move a lot) schoolteacher, he cannot afford to have many belongings, and as a schoolteacher in rural areas, his salary would be so low that he could not afford to buy anything but necessities. In addition, Irving is reminding us that Crane is a New England Yankee from Connecticut, notorious for not wanting to spend any money. This is just a gentle dig at the literary stereotype of someone from New England.