Well, there are two elements to this.

 

First of all, tolerance toward non-Muslims was part of Islam itself. Christians and Jews in the Ottoman Empire and in other Muslim empires were given the right to practice their religion in exchange for the payment of a poll-tax, called cizye in Turkish. These taxpaying non-Muslims were called zimmi. So when historians talk about tolerance toward non-Muslims in the Ottoman Empire, this is typically what they are referring to. This type of tolerance, since it was required by religious law, was accepted and supported by Muslim religious leaders.

 

Interestingly enough, there doesn't seem to have been any Muslim scholarly objection to the practice of devşirme, the gathering of recruits for the various palace corps from among Christian children, despite the fact that this practice directly violated the rights of the zimmi according to Sharia Law. We find many examples of this throughout Ottoman history, whereby the Sharia and Ottoman dynastic law (kanun) are made to compromise rather than the former inherently taking precedence over the latter. Most (though not all) Muslim scholars were accepting of this, recognizing the importance of tradition ('örf) in maintaining stability in the community.

 

Now, in the Late Medieval context in particular, there were some interesting things going on. The fourteenth century Ottoman Empire was not a strictly Sunni state. Anatolia was home to a wide variety of diverse religious traditions, from tribal rituals to Sufism to Shi'ite elements to syncretic saint-worship between Christians and Muslims. As the Ottoman Empire expanded and centralized, it increasingly came to rely for its administration on scholars from the more longstanding Muslim urban centers of Anatolia. Many of these men had received their education in the core Islamic lands of the Middle East, such as at al-Azhar in Cairo. Thus as they worked to introduce more sophisticated administrative practices into the Ottoman Empire, they also brought with them a desire to make the Ottoman Empire conform more closely to orthodox Sunni Islam. Thus, while the empire was always relatively tolerant toward its non-Muslim minorities, from about 1400 on it became increasingly less tolerant toward heterodox Muslims, a trend which reached a climax with the rise of the Shi'ite Safavid Empire in the decade after 1501 and then full-blown warfare between the fully Sunni Ottomans and the so-called "Kızılbaş" heretics, which lasted until the peace of Amasya in 1555.

Kafadar, Cemal. Between Two Worlds: The Construction of the Ottoman State. University of California Press, 1995. Lowry, Heath. The Nature of the Early Ottoman State. Albany: SUNY Press, 2003.