What kind of physical processes are well modeled as poisson processes? For instance, it is easy to understand how Brownian motions appear: when an object receives many small shocks that are roughly i.i.d., the resulting path will look like Brownian motions.
What type of physical processes give rise to Poisson processes?
For instance, with one dimension thought of as time: what kind of physically processes would naturally have the "memoryless property" between spikes?

TL;DR Processes with a probability density of events in time or space that is constant (and independent from what happened before) have an exponential waiting time distribution, and from this you can derive a Poisson distribution for counts

The Poisson process occurs when we count events within some interval of time (or space) and the waiting time between events (or distance between events) is independent of the past history, the time from the last event (or distance till the previous event). That is, the process has memorylessness. For the probability of an event to occur within some infinitesimal small amount of time (or space) it doesn't matter what events happened before.
Many physical processes are like this (or are approximately like this).
For instance. A Geiger counter counts ionizing particles hitting some sensor. If the origin of these ionizing particles is a large ensemble and there is no interaction between particles (or only a negligible one) then at any moment in time you have a the same probability of a particle hitting the sensor independent from how long ago the previous particle was hitting the sensor.
I myself have used mass spectrometers counting aroma molecules. There are millions of these and only some of them enter the spectrometer and the sensor. When some particle hits the sensor then this is not influencing the probability of other particles hitting the sensor. There are still millions minus one left, and these are giving practically the same probability for the next event to occur within some time frame.
Ben mentions in his answer that almost no process is truly a Poisson process. This is correct. The Poisson process is an idealized situation (in the same way almost no practical process is perfectly Gaussian). In the case of the mass spectrometer, I have three deviations:

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*The signal may not be constant in time due to a varying amount of concentration of particles (I still consider this Poisson process but now it is a inhomogeneous process in which the rate coefficient is a function of time and for different times you do not get the same rate).

*There is a little bit of interaction between the particles. The sensor counting the ionized particles can only count particles that are sufficiently far apart. This creates clipping and when particles hit the sensor close to each other then they are counted as a single particle. But, this effect is negligible when the density is not high.

*There is noise added to the signal which is adding a source of variation different from the Poisson process.

In practice, I make sure that the experiment works well and the distribution of the experimental values is close to a Poisson process. I even used it as a test (checking whether variance and mean of the signal are equal) to control whether the machine is operating well and is not in the regime of clipping or having too low signal to noise.
