The main question:
If 1 pint of paint is needed to paint a statue 6 ft. high, then the number of pints it will take to paint (to the same thickness) 540 statues similar to the original but only 1 ft. high, is
(A) 90
(B) 72
(C) 45
(D) 30
(E) 15
The matter I can not understand:
In the answer it is said that the number of pints is proportional to the square of height. Can anyone explain me why it is proportional to the square of height?
Consider a small square area, say of side $x$, of any portion of a given figure. With a figure that is similar, this area would be stretched or compressed by a certain factor, say $s$, in both the width & height. Since the area of the original square is the product of its side lengths, $x^2$, the area of the similar square area would be $sx \times sx = s^2 x^2$. Adding up all these small areas gives that the new surface area would be proportional to original surface area by the square of the scaling factor. In your particular case it would be $s = \frac{1}{6}$, i.e., the new height of $1$ divided by the original height of $6$, so (as Rhys Hughes has stated in his answer) the proportional area value would be the square of it, i.e., $\frac{1}{36}$.
For example, consider an original height of $h_0$ and an area of $A_0$. Then for a new height $h$, the area would be $A = A_0\left(\frac{h}{h_0}\right)^2 = \left(\frac{A_0}{h_0^2}\right)h^2$. As $A_0$ and $h_0$ are known, constant values, we can let $k = \frac{A_0}{h_0^2}$ be a constant, giving that
$$A = kh^2 \tag{1}\label{eq1}$$
showing that the area is proportional (with a constant of proportionality of $k$ as given) to the square of the height. Similarly, as the amount of paint is proportional to the area, we get
$$P = kh^2 \tag{21}\label{eq2}$$
where $k = \frac{P_0}{h_0^2}$ and $P_0$ is the amount of paint required for the original object.
Note the number of pints is also proportional to the square of the width, but as only the heights are provided, this alternate fact is not needed or used.