I think your question is about hazard rates. The connection between Poisson and Exponential is also addressed here in detail: Relationship between poisson and exponential distribution
The typical intuitive example is this: suppose we are interested in finding the probability distribution of the number of cars that will drive through a particular intersection during a period of one hour. One strategy would be to divide this one-hour period into many, very small periods: periods that are so small, that the probability that more than one car will cross the intersection during each period is negligible (i.e. zero). These $n$ very small periods can now be thought of as containing Bernoulli random variables $x$, with $p$ being the probability of observing a car, and $(1-p)$ the probability of not observing a car. The sum of these Bernoulli random variables would then give us the number of cars for an hour and, assuming they are i.i.d., this sum is Binomial distributed. But, how can we choose an appropriate $n$? With $n=\infty$, that Binomial distribution actually becomes the Poisson distribution.
Finally, here is the intuition you are looking for. If the number of cars you observe each hour (the number of events in each time period $t$) is Poisson distributed, then the time that passes between each event of observing a car has an Exponential distribution. A process like this, where the counts are Poisson and the durations are Exponential, has a constant hazard rate.
Hazard is the instantaneous probability of observing a car right now, given that we haven't observed a car for a while. How long a while? If the hazard is constant, it doesn't matter. It's counterintuitive, but constant hazard in this example means that the probability of observing a car is the same whether I've been waiting at the intersection 1 hour or 24 hours.
We can of course also model positive duration dependence (the longer I wait, the greater the probability of finally observing a car) and negative duration dependence (a patient that hasn't died 2 months after surgery has a lower probability of dying from complications than a patient that hasn't died 2 hours after surgery). A popular choice for this is the Weibull distribution, of which the Exponential distribution is a special case.