Mozart lived in Modern Europe (died 1791, soon after the French Revolution); not in medieval, obscure times. Mozart's music was sold (print music sheets), adapted, re-arranged, played by memory, played at home, on the streets, on theaters, etc., during his lifetime enough so that most people in Central European countries would have heard it. Most people would have listened to enough Mozart to recognize his personal style, top melodies, etc.
Music prints were generally available (see: earliest Mozart editions) and although expensive, people copied them themselves at home very cheaply (also, cheap copies were sold). Playing the violin was quite popular, and playing the piano was to become very popular very soon (think 1810). Small string orchestras were also very common in all the German-speaking area to play symphonies and such.
Mozart was a famous child prodigy touring courts, and then, later in life he become "a normal composer". Although famous (think like a Kardashian), he often struggled financially, just like any musician, because at the time, music was not yet an industry and was not perceived as something "useful" people paid for.