So this peasant in Bulgaria named Ivaylo says he's tired of all these Mongols and gets together a peasant army. How was this roughshod force able to repel the Mongols when so many others have failed?
Actually, others had not failed. 30 years earlier Ivan Ashen II had also driven the Mongols out of Bulgaria. The mountainous terrain in the central spine of the country makes it not the best terrain for the Mongol way of warfare, and totally unsuitable for their pastoral way of life. So the country was really only good for them for occasional raiding.
The reason the Mongols were back was because after Ivan the monarchy was weak, and had resorted to paying the Mongols tribute rather than fighting them. This worked OK for the upper classes, but obviously not so great for the peasantry.
He didn't "repel" anything and I doubt Ivaylo defeated any significant Mongolian force. What it says in an article "Les Mongols dans les Balkans" by Gaston Cahen in Revue Historique (T. 146, Fasc. 1, 1924, pp. 55-59) is the following:
La Bulgarie, apres les assenides et dans les dernieres annees de constantin, etait partagee entre les factions rivales des boyars. le tsar, malade, fut defait et tue en 1277 par un simple berger, ivailo, qui se posa en adversaire des tatares et occupa la capitale, Tirnovo (1277).
Bulgaria, after the Assenides and in the last years of Constantine, was shared between rival factions of the boyars. The Tsar, sick, was killed in 1277 and usurped by a simple shepherd, Ivailo, who posed as an opponent of Tatars and occupied the capital Tirnovo (1277).
From this it appears that upper Bulgaria was not occupied by the Mongols at all and was, at most, a vassal state. In the confusion of a succession crisis caused by the illness of Constantine Asen, Ivailo appeared pretending to oppose the Tatars (Mongolians). However, once he was admitted to the city, he killed the ailing Tsar and seized the throne by force.