How the Poisson distribution is used in regression? I am trying to intuitively understand the Poisson regressions, but I don't get:

*

*why the increasing variance along with the mean is relevant?

*how each of these three theoretical Poisson distributions was shown here, i.e. using each Y value as a mean=variance?

So far, my understanding is that for each Y a theoretical distribution is drawn, and then the best curve is fitted to maximize the cumulative probabilities (maximum likelihood?)

 A: 
why the increasing variance along with the mean is relevant?

Remember that the variance of a Poisson random variable is equal to its mean.  If the mean increases, so too does the variance.

how each of these three theoretical Poisson distributions was shown here, i.e. using each Y value as a mean=variance?

Regression is all about the conditional distribution of the outcome.  If you fix $x$ and observe $y$ a bunch of times, and then look at the histogram of those $y$, then that histogram should be Poisson.
A: *

*For $Y\sim Poisson(\lambda)$, $\mathbb E\big[Y\big] = \lambda$ and $\text{Var}(Y) = \lambda$. If the mean of the conditional distribution increases, the variance of the conditional distribution must increase, too. If there is no such increase, too much increase, or too little increase, then Poisson would appear not to be the right conditional distribution. Negative binomial could be an alternative.


*In your plot, it looks like $\hat y(1.75) = 2.5$, $\hat y(3) = 4.8$, and $\hat y(4.7) = 10$. That is, the conditional distributions are $\big(Y\vert X = 1.75\big) \sim Poisson(2.5)$,
$\big(Y\vert X = 3\big) \sim Poisson(4.8)$,
and $\big(Y\vert X = 4.7\big) \sim Poisson(10)$. The plots shows sideways plots of those distributions, positioned at the appropriate $x$-positions.
For that regression curve, you posit that $\log(\lambda_i) = \beta_0 + \beta_1 x_i$, equivalent to $\lambda_i = \exp(\beta_0 + \beta_1x_i)$, and you find the maximum likelihood estimator for $\beta_0$ and $\beta_1$, with the curvature explained by the logarithm. (Each $\lambda_i$ is the $\lambda$ parameter of some conditional Poisson distribution (conditional on your predictor variable, $x$).)
