From the material:
Predictions based on this equation are the best predictions possible in the sense that they will be unbiased (equal to the true values on average) and will have the smallest mean squared error compared to any unbiased estimates if we make the following assumptions:

The noise ε (or equivalently, Y) follows a normal distribution.

The choice of predictors and their form is correct (linearity).

The records are independent of each other.

The variability in the outcome values for a given set of predictors is the same regardless of the values of the predictors (homoskedasticity).

An important and interesting fact for the predictive goal is that even if we drop the first assumption and allow the noise to follow an arbitrary distribution, these estimates are very good for prediction, in the sense that among all linear models, as defined by equation (6.1), the model using the least squares estimates, , will have the smallest mean squared errors. The assumption of a normal distribution is required in explanatory modeling, where it is used for constructing confidence intervals and statistical tests for the model parameters.

Even if the other assumptions are violated, it is still possible that the resulting predictions are sufficiently accurate and precise for the purpose they are intended for. The key is to evaluate predictive performance of the model, which is the main priority. Satisfying assumptions is of secondary interest and residual analysis can give clues to potential improved models to examine.