The initial conquest of Bulgaria was probably more dumb luck than anything else (or, rather, the Ottomans realizing that the Bulgarian kingdom had become weak and that presented a particularly opportune moment to strike). The Ottomans were very good at taking advantage of confusion and disorder.

That said, your question is very prescient: the Turks may have defeated the Bulgarians because they were military weaker, but they held on to the territory because there were advantages to being part of the Ottoman state.

The Ottomans offered stability in a world that was in flux due to the decline of the feudal order in Europe, and instabilities caused by the seemingly never-ending clashes between Venice, Byzantium, and Genoa. The Ottomans brought in military safety, security for internal and external trade, and allowed the Balkan kingdoms and principalities to be plugged into the larger requisitioning system and trading system within the Ottoman empire, which provided cushioning against famine or shortages.

The Christian kings and princes of Balkan states often became vassals of the Ottoman sultan (this was largely the same way that it worked with the Byzantine or Holy Roman emperors). As in Europe, family politics rules the day: Ottoman sultans and their sons married sisters and daughters of princes and kings; in return for acknowledging the sultan as their superior and pledging loyalty, the Balkan princes or kings or chieftains got to keep their titles and positions. In turn their heirs would get to marry into other noble families, and so on.

Although a bit feudal, this is the same system at heart that continued in the Ottoman empire until the 19th century, the German principalities through unification until World War I (and the same in Austria-Hungary).

The idea that there must have been constant conflict because the Ottomans were Muslim and the Balkans were Christian was a supposition of post-Enlightenment thinkers, but the reality on the ground, especially before, say, 1700, was much more pragmatic. (This is not to say that things were always cheerful and everyone got along -- however, the Ottoman sultans also learned that if they got too authoritarian and demanded too much, their vassals started to rebel). Think of it as a marriage of convenience.

More info can be found in chapter 1 of Donald Quataert's The Ottoman Empire, 1700-1922, 2nd edition (this is one of those books where the edition makes a big difference); this is easily readable; also chapter 2 of Jason Goodwin's Lord's of the Horizons: A History of the Ottoman Empire.

You might also be interested in Mary Neuburger's episode of the 15 Minute History podcast (which can be read): https://15minutehistory.org/podcast/episode-37-the-ottoman-balkans/