Just like any model, you should evaluate the residual diagnostics to better understand how the model is performing - and identify possible elements the model is not capturing.
The comment in the textbook is referring to the way in which forecast distributions are combined.
Assume you have two distributions, X \sim N(\mu_x, \sigma^2_x) and Y \sim N(\mu_y, \sigma^2_y). You can imagine that X is the 1-step forecast from an ARIMA model, and Y is the 1-step forecast from an ETS model.
Now, a forecast combination model may equally weight the forecasts from the ARIMA and ETS model, or in other words Z \sim (X + Y)/2, where Z is the 1-step forecast from the combination model.
If the two forecast distributions X and Z were independent, you could easily say Z \sim N((\mu_x + \mu_y)/2, (\sigma^2_x + \sigma^2_y)/4). However, X and Z are not independent, they are forecasts of the same thing! Instead we need to take into the correlation/covariance between the residuals of the combined models. This gives \sigma^2_z = \sigma^2_x + \sigma^2_y + 2\times \text{cov}(X,Y) instead of \sigma^2_z = \sigma^2_x + \sigma^2_y .
This step of handling the covariance between forecasts is handled automatically and it is not something you should need to worry about. This step does not guarantee the residuals are well behaved in any way - that is up to your choice of model (which by saying 'model' I also include combinations of models).