Properties of the matrix exponential

Power series.
$$e^A e^B = \left(\sum_{k=0}^{\infty} \frac{A^k}{k!}\right)\left(\sum_{l=0}^{\infty} \frac{B^l}{l!}\right) = \sum_{n=0}^{\infty} \sum_{m=0}^{n} \frac{A^{n-m}}{(n-m)!} \frac{B^m}{m!}.$$

Binomial theorem. Since $AB=BA$, $(A+B)^n = \sum_{m=0}^{n} \binom{n}{m} A^{n-m} B^m$. Therefore
$$e^A e^B = \sum_{n=0}^{\infty} \frac{1}{n!} \sum_{m=0}^{n} \binom{n}{m} A^{n-m} B^m = \sum_{n=0}^{\infty} \frac{(A+B)^n}{n!} = e^{A+B}.$$