Choice A is correct. The area $A$, in square meters, of a rectangle is the product of its length $\mathcal{l}$, in meters, and its width $w$, in meters; thus, $A=\mathcal{l}w$. It's given that a rectangle has an area of $63$ square meters and a length of $9$ meters. Substituting $63$ for $A$ and $9$ for $\mathcal{l}$ in the equation $A=\mathcal{l}w$ yields $63=9w$. Dividing both sides of this equation by $9$ yields $7=w$. Therefore, the width, in meters, of the rectangle is $7$.

Choice B is incorrect. This is the difference between the area, in square meters, and the length, in meters, of the rectangle, not the width, in meters, of the rectangle.

Choice C is incorrect. This is the square of the length, in meters, not the width, in meters, of the rectangle.

Choice D is incorrect. This is the product of the area, in square meters, and the length, in meters, of the rectangle, not the width, in meters, of the rectangle.