How do you explain cubic growth of a function

Growth as a quadratic, cubic, quartic, or other fixed-power multiple is referred to generally as polynomial: formulas such as x² or x³+2x+7 or even x⁹⁹+99x⁹⁸+98x⁹⁷+... are all polynomial. (These formulas can be characterized as "when you double your input X, your output will be 4 times (or 9 times, or 64 times) as big.")

Exponential growth occurs when the variable itself appears as the power, e.g. in a formula such as 2^x, for example in a situation where "the population at time t is approximately equal to 1.1^t." (These formulas can be characterized as "when you add 1 to your input X, your output will be doubled / tripled / multiplied by some other factor.")

The mathematical term for a function of a variable that changes with a fixed power (actually the sum of fixed multiples of fixed non-negative powers) is "polynomial." The multipliers are called "coefficients; the powers, "exponents." ("Fixed" here means constant, i.e, the values of the coefficients and the exponents don't change with the value of the variable.

The highest power, called the "degree" or "order" of the function, gives the name of the polynomial:

0 constant
1 linear
2 quadratic
3 cubic
4 quartic
5 quintic

After exponent=5, the names are simply the "n-th power."

I'm not sure what you mean by "explain" the graph. A polynomial function operates by iterated multiplication on the value of the variable, the number of iterations given by its degree. Because multiplying a number between 0 and 1 by itself results in a product that's smaller than the number, the higher the degree of the polynomial, the slower its growth between 0 and 1.

In your graphical example, your polynomials have only one term each, and all their coefficients are 1. These are called "monic monomials." Their values are zero starting at 0 because multiplying 0 by itself any number of times results in 0. The values increase toward one as the variable approaches 1, reaching one at 1. This because multiplying 1 any number of times by itself just results in 1.