Democracy destroys itself because it abuses its right to freedom and equality. Because it teaches its citizens to consider audacity as a right, lawlessness as a freedom, abrasive speech as equality, and anarchy as progress.
Notes from Wikiquote
Οἱ γὰρ κατ᾿ ἐκεῖνον τὸν χρόνον τὴν πόλιν διοικοῦντες κατεστήσαντο πολιτείαν [. . .] οὐδ᾿ ἣ τοῦτον τὸν τρόπον ἐπαίδευε τοὺς πολίτας ὥσθ᾿ ἡγεῖσθαι τὴν μὲν ἀκολασίαν δημοκρατίαν, τὴν δὲ παρανομίαν ἐλευθερίαν, τὴν δὲ παρρησίαν ἰσονομίαν, τὴν δ᾿ ἐξουσίαν τοῦ ταῦτα ποιεῖν εὐδαιμονίαν, ἀλλὰ καὶ μισοῦσα καὶ κολάζουσα τοὺς τοιούτους βελτίους καὶ σωφρονεστέρους ἅπαντας τοὺς πολίτας ἐποίησεν. | Those who directed the state in the time of Solon and Cleisthenes did not establish a polity which [. . .] trained the citizens in such fashion that they looked upon insolence as democracy, lawlessness as liberty, impudence of speech as equality, and licence to do what they pleased as happiness, but rather a polity which detested and punished such men and by so doing made all the citizens better and wiser. | VII.20