Poisson distribution is a limit. Are you familiar with statistical distribuitons?
Flip a coin. The probability of a heads is p, and the probability of a tails is 1-p, where p is the probability of getting heads. This is called the Bernoulli distribution
Now cut the coin into 10 pieces, the probability of getting n pieces with heads landing up is (10 choose n)pn(1-p)10-n. That is (how many ways to get n heads out of all possible coin flips)x(probability of getting any one of those outcomes).
Now cut it into just so many pieces. As many as you can. Provided that you satisfy some conditions about how you do it (specifically that p gets smaller and smaller to the extent that np-> finite number and overlapping conditions) that's gonig to get ever closer to The poisson distribution with parameter p(we usually call it lambda). If you look at some definitions of e, then it comes out clear as day.
The poisson distribution can be seen as a "limiting case" for a binomial distribution where the probability of an outcome gets smaller and smaller.
It seems arbitrary to use, but there are processes where you can conceptualize just this. In the food science lab I worked at, the counts of colonies on an agar plate fit this process really well, so we could model it that wy.