What is the exact purpose of that Next of Kin form?
The Next of Kin form is used if the estate of the registrant is settled.  In that case, the refund is issued in the name of the next of kin.
What should I do if I decide to transfer ownership of my recently deceased mother's car?
You can use PDF [2} MV-843 as the checklist to help you transfer the ownership of the vehicle.