1560s, "a preservative," from noun use of conservatory (adj.) "having the quality of preserving," from Latin conservator "keeper, preserver, defender," agent noun from conservare. Meaning "a place for preserving or carefully keeping anything" is from 1610s, from Latin stem of conservation + -ory. From 1660s as "greenhouse."
Meaning "school of music, for performing arts" is recorded from 1805, from Italian conservatorio or French conservatoire, a place of public instruction and training in some branch of science or the arts, especially music, from Medieval Latin conservatorium. Originally an Italian institution, "hospital for foundlings in which musical education was given;" it was picked up by the French after the Revolution. The Italian word is attested in English from 1771.