Refers to tea etiquette where the pinky finger is extended while drinking. The phrase is used sarcastically calling someone less than classy.

From Tea Etiquette: where the pinky finger is extended while drinking.

When you are about to do some thing classy, expensive, or are exaggerating the value of something.

to act snooty,to put on aires, to pretend to be high class, exaggerating one's own importance.