Hydraulically tells you how something is moved or worked: by the pressure of a liquid in pipes. “The tailgate opens hydraulically” means the liquid does the lifting, not a person and not a simple spring. You will see it in manuals and on product pages more than in chat. It is hydraulic with -ally on the end, the usual way English turns this kind of adjective into an adverb. Same real picture as hydraulic: the liquid is often oil, even though the old root points to water.