Why does high EDTA concentration cause swelling of platelets?
As it's reported in an old 1968-article by J. G. White:


  Chelation of membrane calcium by EDTA appears to cause marked
  irregularities in the platelet wall and massive swelling of the
  canalicular system. Swelling of internal channals is the dominant
  alteration in EDTA platelets, and this change appears ultimately to be
  related to the breakdown of platelet granules.


Many articles followed the one reported above, but it seems that the main cause of platelets swelling in presence of EDTA is the reduction of calcium levels that causes a sort of platelet activation.

This acivation could also be linked to direct EDTA action as reported in this article by Golański et al.:


  The influence of EDTA on platelet membrane dynamics closely mimicked
  the alterations induced upon the interaction of fibrinogen with
  platelet GPIIb-IIIa. Thus, the EDTAK2-induced platelet activation
  might result from an interference with platelet membrane protein
  structure and conformation and possibly relate to an “unspecific”
  trigerring of a signal transduction pathway.


Anyway in normal conditions the laboratory platelet volume quantification using EDTA tubes seems not to be affected if executed in 1 hour since the collection (as reported in this article by Dastjerdi et al.).