Why are the standardized residuals of a Poisson distributed random variable approximately normally distributed? I'm currently reading through this GLM textbook, and have come across this assertion on page 24 that I can't quite wrap my head around

The author claims if the expected value $\theta_i=E[Y_i]$ of Poisson distributed data from independent Random Variables $Y_i$. How is this true? Is there an assumption they are making? I initially thought it may have something to do with CLT, but I'm not completely sure.

As MikePhifer notes correctly, the caveat is that the expected mean must be "not too small" (in practice, I would say at least > 10 or so). In this case, the Poisson is approximately normal, and so are the residuals; however, if that is the case, you might as well fit a log-transformed lm.
In all cases where you really need a Poisson GLM, standardised residuals will be non-normal. However, you can use the idea of quantile residuals, e.g. the DHARMa package in R, to transform to uniformity (or other distributions, including normality, see options in residuals.DHARMa)
