We establish a Perron-Frobenius type theorem for products of stationary and ergodic nonnegative random matrices, which provides a precise asymptotic description of their coefficients, and reveals a close connection with the ergodic theory for stationary and ergodic sequences of real random variables. As applications, we improve upon existing laws of large numbers for products of nonnegative random matrices by relaxing moment conditions. We then derive limit theorems for a multi-type branching process $(Z_n)$ in a stationary and ergodic environment, offering a precise description of the growth rate of the population size. In particular, we establish a Kesten-Stigum type theorem for the scalar product $\langle Z_n, y\rangle $ with any nonnegative vector $y$, revealing a key link between the branching process and products of random matrices. We also prove the convergence of the direction $Z_n/ \|Z_n\|$, a complement to the Kesten-Stigum type theorem, and establish new laws of large numbers and central limit theorems for $\langle Z_n, y\rangle$.