"This week, we examined the simplest theory of spatial criminology – that crimes are distributed at random. When spatial criminologists want to test observed data for randomness, they test that data’s distribution against the Poisson distribution and look for a statistically significant difference. What is the null hypothesis in this situation? What conclusion would we draw from rejecting the null hypothesis?"
If the occurrence of events is homogeneous and independent then the number of events per unit of exposure (e.g. area) will have a Poisson distribution, so you could test for a null of that particular kind of randomness by doing a goodness of fit test of a Poisson distribution.
You can then phrase the null either as a null of the process being random (a Poisson process) or as having a Poisson distribution.
If your null is of a non-homogeneous process then things become more complicated than that.
I would add that this seems a highly implausible model and I see no good reason to be testing for it.