Say you move to a country where everyone cooks with lots of hot peppers. At first the food scalds your tongue, but over time you adapt — you change in a way that allows you to deal with the new circumstances.
Adapt comes from the ancient word ap, which means "take" or "grasp." Ap is even older than Latin — it comes from a lost language that was spoken by the common ancestors of modern-day Indians and Europeans and has since been reconstructed by linguists, who named it the Proto-Indo-European language, or PIE. So what does adapt have to do with grasp? If you adapt to, say, a new country, it's as though you're grabbing hold of its strange, slippery customs.