The underlying mathematics of all types of regression assume that there are no patterns in the residuals. That is, they assume that the difference between the observed and predicted values of the model reflect only random noise. The reason that they make such an assumption is that if the errors of the model reflect something other than random noise, then the model is failing to account for something important.
The various tests that exist of the assumptions of regression mainly relate to looking for different types of patterns in the residuals.
See How Does Linear Regression Work? for more detail.