To “wax” means to grow or to become. If some is waxing intellectual it means that they are assuming the role of an intellectual.
Waxing and waning refer to something getting larger and smaller, respectively, like the waxing and waning of the moon. If someone is waxing intellectual, then his behavior is “increasing” toward an intellectual way of talking or writing. Similarly, if one “waxed poetic,” then their phrasing would have taken a poetic turn.
someone who thinks they’re smart and tries to show it by talking too much and using big words and trying to impress people and get their praise