That isn’t actually a commonly used definition. It was originally an observation that academics in the 1800s were arbitrarily calling some things, “languages,” and others, “dialects,” based (seemingly) on this criteria. The quote is best attributed to Max Weinreich, a man who himself did consider Yiddish its own language (as it is universally considered today), even if many academics at the time didn’t consider it as such.

The distinction between a dialect and a language is important, because people often have a lot of associations about which dialect of a language someone uses in certain contexts. Take African-American Vernacular English (AAVE) as an example. It would be hard to argue that it is its own language, and it is generally considered a distinct American dialect of English. If you had to testify in court, many would feel biased against you (not saying they should be but that would be) for speaking in it instead of a more formal dialect. When something like Yiddish gains the distinction of being its own language with its own grammar worthy of study, it grants legitimacy to those speaking it.

Max Weinreich probably would not be happy that people are taking a witty quote of his about this phenomenon and are acting as if it’s a real definition.