According to the online dictionary below, the word is of Latin origin, "cura" = care, trouble, concern. It then pased into Old French, and then from there into the English language. Makes sense, as it is similar to our modern English word "care'. We also have the word "curator", of a museum, for example.It’s also used in non food ways like curing rubber.
Maybe it was to do with taking care and eliminating a problem. Meat would’ve been a problem if it was not cured?
cure | Etymology, origin and meaning of cure by etymonline
Not mentioned in the online dictionary, but of interest: modern Greek has the word "κουράζω" (curazo) = tire (can mean to trouble someone), become tired. Ancient Greek has επικουρέω (epicureo) modern Greek epicuro = to help.