Why does brining lead to firmer salmon?
Excellent question. The salt, through osmosis, takes humidity out of the meat. Less moisture leads to a denser product.

About meat:


  The main structural component of the muscle fibers in meat is
  myofibril, which is itself composed of thick and thin filaments.
  Higher-than-normal levels of salt cause these filaments to swell and
  separate from one another and depolymerize, or break down. This
  process makes meat more tender.
  
  For tenderization, salt is usually added to meat via a brine solution
  comprised of salt, water, and sometimes other flavoring compounds.
  Harold McGee describes the effect of brining in his book "On Food and
  Cooking"


Link

I hope that answers your question.

Edit I only answered half the question.

Link to a more detailed explanation about osmosis.