For a random variable $X \sim Po(λ)$ we have that $\mathbb{P}(X = x) = \frac{λ^x e^{-λ}}{x!}$
With a little unwrapping/rearranging this turns into $\mathbb{P}(X = x) = \frac{\frac{λ^x}{x!}}{\sum_{k \geq 0}{\frac{\lambda^k}{k!}}}$
More clearly if we let $f(x) ≔ \frac{λ^x}{x!}$ then we have $\mathbb{P}(X = x) = \frac{f(x)}{\sum_{k }f(k)}$
I'm wondering if there is some way of thinking about a Poisson random variable that makes it clear why each outcome of the random variable corresponds precisely to one of these $f$ terms in the expansion for $e^{-\lambda}$ and/or something which makes it clear where this ratio structure of $f(x)$ to $\sum_kf(k)$ comes from.
The closest I've found is the accepted answer to Intuitive explanation of Poisson distribution, but after thinking about it I still feel like I'm missing a few pieces. I don't quite see where the integral is coming from, and how it relates to a Poisson distribution, and I don't quite see why it's justified that you can sum these integrals over the naturals to build a sample space (although I'm assuming part of the reason I don't understand why you can do this is because I don't really understand where the integrals are coming from)
I'm not attached to the idea of understanding the above explanation so if it's easier to answer my question without referencing it feel free to do so.
