The area of the base is $x^2$ square meters (since it is a square with side length of $x$ meters). There are 4 sides to this box, and each side will be a rectangle with a width of $x$ meters and a height of $y$ meters (the height of the box). The volume of the box is $x^2y$, the area of the base times the height (this is just the volume of a rectangular prism).
Thus, there is $x^2$ square meters of base material needed, and $4xy$ square meters of side material needed. Given the prices, this means that the total price of building the box is $4(x^2)+5(4xy)=4x^2+20xy\;\;\text{ dollars.}$ Assuming that the problem means that you are supposed to spend exactly 60 dollars on the box, this means that $4x^2+20xy=60$ This constraint will let you determine the volume of the box, $x^2y$, even if you only know the value of $x$ (or, if you only know the value of $y$). Suppose I tell you that $x=1$. Then you know that $4+20y=60$ $y=\frac{56}{20}=\frac{14}{5}$ Thus, the volume of the box in this case is $x^2y=1^2\cdot\frac{14}{5}\text{ cubic meters}.$ You can also solve for $x$, if you know $y$, though it is a little trickier. This means that you will also be able to find the value of $x^2y$ in this case.
The problem is asking you to do this in general - given one of the variables, say $x$, solve for the volume of the box you will get, using the fact that you spend exactly 60 dollars.