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.

John, good question and welcome (I see both of us are new here). I think it might be related to the Poisson Normal approximation (ballpark est.) with the general rule of thumb of if Poisson mean count (rate) is 5 or more.
Normal Mean ~ Poisson expected Count, and Normal Standard Deviation (sigma) ~ Sqrt(Poisson Expected Count). Combine this with a standard normal Z calculation (Z=(x-mean)/sigma) and you get the residual Ri equation.
Here is a Monte Carlo for Ri with Poisson Rates: 0.1, 1, 5, and 20. Note that 0.1 and 1 are not well modeled by this, 5 is only roughly modeled and for Poisson(20) the model is better/ reasonable. Probability (CDF) Plot Assuming Normal Distribution for Ri 
