Looking at the PMF of the Poisson distribution:
$$P(X=x) = \frac{e^{-\lambda}\lambda^x}{x!} =e^{-\lambda} \frac{\lambda^x}{x!}$$
The $\frac{\lambda^x}{x!}$ is the $x$-th term in the Taylor expansion of the exponential function. And the $e^{-\lambda}$ normalizes those terms and makes them sum to $1$.
Is there an intuitive reason why the terms of the Poisson PMF must be proportional to the Taylor expansion of the exponential function? Perhaps through the Poisson process?