Fundamental theorem of finite Markov chains

Existence and uniqueness. Since $P$ is row-stochastic, the all-ones vector is a right eigenvector with eigenvalue 1, so 1 is also an eigenvalue of $P$ (a matrix and its transpose share eigenvalues), and there is a corresponding left eigenvector $\pi$ with $\pi P=\pi$. Irreducibility means $P$ is the transition matrix of a strongly connected graph, so the Perron–Frobenius theorem applies: the eigenvalue 1 is simple (multiplicity one) and its eigenvector can be chosen strictly positive, giving uniqueness after normalizing so entries sum to 1.

Convergence. Aperiodicity plus irreducibility means all other eigenvalues $\lambda$ of $P$ satisfy $|\lambda|<1$ (this is the part periodicity would break: a periodic chain has extra eigenvalues exactly on the unit circle, such as $-1$, which never die out). Writing an arbitrary starting distribution $\mu_0$ in the eigenbasis of $P$, the component along $\pi$ (eigenvalue 1) survives unchanged forever, while every other component is multiplied by $\lambda^n$ at step $n$ and shrinks geometrically to zero.

Combining the two steps, $\mu_0 P^n$ converges to the eigenvalue-1 component alone, which is exactly $\pi$, and since this holds for every starting distribution $\mu_0$ (including each point mass, i.e. each row of the identity), every row of $P^n$ converges to $\pi$ as claimed.