Danny Gorenstein, in his book Finite Simple Groups, An Introduction to their Classification discusses a very significant application of Frobenius' theorem (see p. 95). He is interested in the following central problem in the Classification of Finite Simple Groups:

Let $G$ be a simple group in which the structure of the centralizer of an involution is given. Determine the order of $G$.

This problem naturally splits into different cases, the most difficult of which concerns the case where $G$ has a single conjugacy class of involutions. One proceeds by studying the $2$-local structure of $G$, then the $p$-local structure, for those primes $p$ dividing $|X|$ where $X$ is the given involution centralizer.

Gorenstein remarks that

With this information, one can now obtain a congruence for the order of $G$ with the aid of Sylow's theorem and a result of Frobenius.

The `result of Frobenius' he refers to is the one in this question. I'm unqualified to write more so I recommend Gorenstein's book for more details (on this and anything else to do with CFSG).

One final remark: Gorenstein notes in a footnote that Hall's book The Theory of Groups contains a strengthening of Frobenius' result (see Theorem 9.1.1, I'm not sure if this is the same generalization mentioned by Anton above):

If $X$ is a conjugacy class of elements of the group $G$, then for any positive integer $n$, the number of solutions in $G$ of the equation $x^n=c, c\in C$, is a multiple of ${\rm gcd}(|C|n, |G|)$.

Danny Gorenstein, in his book Finite Simple Groups, An Introduction to their Classification discusses a very significant application of Frobenius' theorem (see p. 95). He is interested in the following central problem in the Classification of Finite Simple Groups:

Let $G$ be a simple group in which the structure of the centralizer of an involution is given. Determine the order of $G$.

This problem naturally splits into different cases, the most difficult of which concerns the case where $G$ has a single conjugacy class of involutions. One proceeds by studying the $2$-local structure of $G$, then the $p$-local structure, for those primes $p$ dividing $|X|$ where $X$ is the given involution centralizer.

Gorenstein remarks that

With this information, one can now obtain a congruence for the order of $G$ with the aid of Sylow's theorem and a result of Frobenius.

The `result of Frobenius' he refers to is the one in this question. I'm unqualified to write more so I recommend Gorenstein's book for more details (on this and anything else to do with CFSG).

One final remark: Gorenstein notes in a footnote that Hall's book The Theory of Groups contains a strengthening of Frobenius' result (see Theorem 9.1.1, I'm not sure if this is the same generalization mentioned by Anton above):

If $C$ is a conjugacy class of elements of a finite group $G$, then for any positive integer $n$, the number of solutions in $G$ of the equation $x^n=c, c\in C$, is a multiple of ${\rm gcd}(|C|n, |G|)$.