The practical effect of the statue is to reduce the recordation tax. 

Here in Virginia the recordation taxes are collected based on the amount of consideration recited in the deed of trust (mortgage). Where the borrower has refinanced with the same lender, the calculation of the recordation tax is based on the difference between the amount of the new loan and the old loan -  theory being that the taxes have already been collected on the amount of the old loan, and having to pay based on the full amount of the new loan would be in effect, a double tax. 

The refinancing affidavit is proof of the amounts used to make the calculation for the tax, and proof that the new loan is with the same lender.