This symbol is called a “tilde.” The name comes from Portuguese and Spanish, and from the Latin word "titulus." In written English, it is used to mean “approximately.” Example: “The distance is ~30 miles.”
In some languages it serves as a diacritical mark above a letter to designate a change in pronunciation, similar to the "umlaut" in German. It had a similar function in ancient Greek and Medieval Latin and it also plays a role in many modern languages as well. The tilde performs multiple duties and has an assortment of meanings in mathematics, electronics, computing, and even juggling notation.
The use of a tilde to attribute the source of a quote is a matter of personal choice.