To luxuriate is to enjoy yourself extravagantly or to an extreme degree. Luxuriate can also mean to thrive, like a plant that grows wildly.
A luxury is something expensive and unnecessary, like eating at a fancy restaurant. To luxuriate is to enjoy yourself in a similar way, not necessarily by spending a lot, but by enjoying something to the fullest. Lying in the sun for hours is luxuriating. Reading in the bookstore for hours is luxuriating. Also, luxuriating is a type of prosperous growth. A banana tree that yields a lot of bananas is luxuriating: it's thriving and growing.