It doesn't. A statistician would call the three covariates 'nuisance variables', which are things that you think (know) will have an effect on at least some of the genes, and so you need to account for them in your model.

In your case you said you have paired samples. So if one patient has a much higher level of expression for a given gene than the other patients, you don't necessarily care about that fact. Instead, you want to know how much the treatment affects the gene expression after accounting for any patient-specific differences. And you account for any between-patient differences by adding a patient-level blocking factor to your model, which estimates the mean expression for that patient, and removes that patient-level expression, to give you a cleaner signal.

Put another way, this is just simple algebra. You are saying, that for Gene X, the expression for patient 1 is

Gene_X_expression = patient_effect + treatment_effect + covariate1_effect + covariate2_effect + covariate3_effect

Which is the same as saying that

treatment_effect = Gene_X_expression - patient_effect - covariate1_effect - covariate2_effect - covariate3_effect

So the treatment effect is estimated from the expression data, after subtracting out all these other things that might affect the gene expression, but are not of interest to you. It's a little more complicated than that, but that's the basic idea.