What is the implication of Perron Frobenius Theorem?

The Perron-Frobenius theorem is used in Google's Pagerank. It's one of the things that make sure that the algorithm works. Here it's explained why the the theorem is useful, you have a lot of information with easy explanations on Google.

For example, there are important applications in the theory of finite Markov chains. Among other things, it implies that a Markov chain with strictly positive transition probabilities has a unique stationary distribution.

First of all, the conclusion is more interesting if you also try an example where the matrix entries are still real, but not positive. For example, if $A$ is the $2\times2$ rotation matrix $$A=\pmatrix{\cos\theta&\sin\theta\\ -\sin\theta&\cos\theta}$$ then you can check that the eigenvectors are no longer real, and there's no longer a unique eigenvalue of maximum modulus: the two eigenvalues are $e^{i\theta}$ and $e^{-i\theta}$.

I second Robert Israel's nomination of Markov chains as a great application. I'll add the following, though. (I'll remark that you don't actually need the matrix to have positive entries - just for some power of it to have positive entries.) Suppose you have a finite connected graph (or strongly connected digraph) such that the gcd of the cycle lengths is $1$. Then if $A$ is the adjacency matrix for the graph, some power $A^r$ will have positive entries so Perron-Frobenius applies. Since the entries of $A^n$ count paths in the graph, we conclude that for any pair of vertices in the graph, the number of paths of length $n$ between them is $c\lambda_1^n(1+o(1))$, where $\lambda_1$ is the Perron-Frobenius eigenvalue and $c$ is a computable constant. Applications of this particular consequence of Perron-Frobenius include asymptotic growth rate results for "regular languages," which are defined in terms of paths on graphs.

In particular, there is a cute formula giving the $n$-th Fibonacci number as $$F_n=\left\langle \frac1{\sqrt5}\phi^n\right\rangle$$ where $\phi$ is the "golden ratio" $(1+\sqrt5)/2$ and $\langle\cdot\rangle$ denotes "closest integer." This formula, and many others like it, become less surprising once you check that $$\pmatrix{1&1\\1&0}^n=\pmatrix{F_{n+1}&F_n\\ F_n&F_{n-1}}$$ and note that $\phi$ is the Perron-Frobenius eigenvalue (and the other eigenvalue is smaller than $1$ in absolute value.)