Burris Ewell does not know how to spell his name, because none of the Ewells are literate.
The Ewells are a family that live near the dump and are extremely poor. Lee introduces them in this early chapter to foreshadow their significance later on. The fact that they are so poor and illiterate will become crucial. Also, the fact that the teacher, Miss Caroline, does not know that the Ewells are poor is important. It allows the reader to be introduced to them.
When Miss Caroline does not know that Burris Ewell cannot spell, it is because she is new to town. If she wasn’t, she would know that the Ewells only come on the first day of school.
"They come first day every year and then leave. The truant lady gets 'em here 'cause she threatens 'em with the sheriff, … She reckons she's carried out the law just gettin' their names on the roll and runnin' 'em here the first day….” (Ch. 3)
The Ewells live life on their own terms. They exist on the outskirts of society not just physically, but in terms of social conventions. Not going to school is just one example. The fact that little Buriss does not know his letters should come as no surprise. Later, when Mayella accuses Tom Robinson of rape, we learn all kinds of sordid details about the sad existence of this family.
The exception to the going to school rule foreshadows the Ewells other run-ins with the law. The Ewells are a rough crowd. There are so many of them that poor Mayella cannot keep up. She has to watch the children all by herself because her mother is dead, and her father is drunk most of the time. She leads a lonely existence. She likely never went to school either.
Nobody was quite sure how many children were on the place. Some
people said six, others said nine; there were always several
dirty-faced ones at the windows when anyone passed by. (Ch. 17)
When Lee introduces the concept of the Ewells in this chapter, who never bathe and never go to school, it foreshadows Mayella's troubles and the situation that led to the encounter with Tom Robinson. It also partly explains why Bob Ewell reacted the way he did, in beating up Mayella and accusing Tom.
Buriss Ewell can't read, and never will learn. He is one of a brood of uncared for children, raised by a lonely young woman who is trapped in a situation she did not create and cannot control. Issues of class and race interweave in this book, and it all starts here, when Burris Ewell and Scout explain it to Miss Caroline.