The Sensodyne marketing includes an explanation, some of which includes a reference.
They claim that the products containing Potassium Nitrate work because the potassium stops the nerves from firing, citing Nagata, T. et al, Clinical evaluation of a potassium nitrate dentrific for the treatment of dentinal hypersensitivity, J Clin Periodontal. 1994;21:217-221.
They claim the products containing Strontium Chloride work by blocking the exposed tubules within the dentinal tissue, that stops fluid flow that triggers pain. They do not provide references on their web-site to support that claim.