Let $X\sim Exp(\lambda)$, then survival fimction $\mathbf{P}[X\ge x] = e^{-\lambda x}$.
$X$ is interpreted as a waiting time. $X$ is memory-less; the key-property of exponential random variables.
The rest of the story is that the inter-arrival times of a Poisson process are independent of each other, and are exponentially distributed.
In fact, there is an equivalence between saying that events arrive according to such i.i.d. exponentially distributed inter-arrival times and saying that the arrival times are the jump times of a Poisson process.
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