The fundamental role of aryl diazonium salts for post-synthesis selectivity of carbon nanotubes is investigated using extensive electronic-structure calculations. The resulting understanding for diazonium-salt-based selective separation of conducting and semiconducting carbon nanotubes shows how the primary contribution comes from the interplay between the intrinsic electronic structure of the carbon nanotubes and that of the anion of the salt. We demonstrate how the electronic-transport properties change upon the formation of charge-transfer complexes and upon their conversion into covalently attached functional groups. The results are found to correlate well with experiments and provide for the first time an atomistic description for diazonium-salt-based chemical separation of carbon nanotubes.