What is the meaning of this formula regarding Poisson distribution? I'm new to statistics and probability, and I'm dealing with a question about the Poisson distribution. The following is given :

The question is what is the statistical model, and this is the answer for the question:

I dont have the full solution so I'm having some trouble trying to make sense of this. Why is this the statistical model?
The parameter for this distribution is λ (lambda), so they just put this e^BX phrase in place of λ right? still, why is there y in the model? it is not clear to me.

If $Y\sim Poisson(\lambda)$ had a Poisson distribution with parameter $\lambda$, then you could say it probability mass function was $$\mathbb P(Y=y) = \frac{\lambda^y}{y!} e^{-\lambda}=(y!)^{-1} \lambda^y e^{-\lambda}$$ with $y$ being a non-negative integer.
Your expression is the same when  $\lambda=e^{\beta x}$.  Note that  $\left(e^{\beta x}\right)^y=e^{\beta y x}$,
