The phrase in question operates primarily as a suggestive idiom, often used humorously or flirtatiously. It implies a playful offer of a “ride” (typically a kiss or other form of physical affection) in exchange for, or as a result of, the presence of a mustache on the male individual.
The appeal, if any, derives from the underlying humor, the implied flirtation, and the cultural associations of mustaches themselves. Throughout history, mustaches have carried connotations ranging from virility and sophistication to rebelliousness and artistic flair. The phrase leverages these established associations to create a lighthearted, albeit potentially suggestive, interaction. The “free” component adds an element of perceived value or reward, further incentivizing the interaction (in jest, of course).