Chcham: The Khazers were a group in central Asia, and some of their ruling class converted in themiddle ages. When their country broke up, the Jews there moved to Sefardi countries (people traced some of them). In all fairness, except for some elites, the bulk of the Khazer Jews were probably Jewish refugees from the Roman (Byzantime) Empire. At the time the Khahazar state was destroyed, Europe was rather backwards and there was nothing to attract them since they had the option of moving to Arab territories (and note that the Khazar Jews had considered themselves an extension of the Jews in Bavel- which is where they went to learn and to ask shalohs).
The idea of Ashkenazim being descended from Khazers was dreamed up by some frei Jews for several reasons: 1) so they could disassociate themselves from klal Yisrael without feeling guilty about cutting themsevles off from the people; 2) so they could tell the Russians (in the Czarist period) that they were descended from those who the Russian believed killed their diety; 3) so they could tell the Nazis they really were Aryans, and as such shouldn’t be persecuted by.