How do you know there is 9 square feet in a yard?

I don't since a square foot is a measure of area whereas a yard is a measure of length.

However, in 1 square yard there are 9 square feet:

1 yard = 3 feet

1 square yard = 1 yard x 1 yard (the area of a square with sides 1 yard)

= 3 feet x 3 feet

= 9 square feet