Summary: Poisson regression is a generalized linear model for count data.
To model a dataset that is generated from a [[Poisson distribution]] Poisson Process , we only need to model the mean $\mu$ as it is the only parameters. The simplest model we can have for some given features $X$ is a linear model. However, for count data, the effects of the predictors are often multiplicative. The next simplest model we can have is
$$ \mu = \exp\left(\beta X\right). $$
The $\exp$ makes sure that the mean is positive as this is required for count data.