The link between the Poisson process and the distribution of bus arrival times, is firstly that it's a good idealized example for understanding and reasoning about Poisson and exponential distribution, and secondly that the Poisson process somehow seems to be an accurate representation of our
mental model of how waiting for a bus works.
If buses are scheduled to come quite frequently (and don't disappear, so they do arrive with that frequency on average), but the variance in their arrival times is a lot bigger than the average gap between them, then the distribution is actually quite close to Poisson.