Part of the reason why the Ottomans gave Jews shelter was because of the protection given to the dhimmi, or People of the Book (Jews and Christians), by Muslim rulers, which was established by the Pact of Umar. As long as they were submitting to Muslim political rule, Christians and Jews were allowed to continue practicing their faith in exchange for a poll tax called the Jizya. Although allowed to practice their faith, there were restrictions, such as: not being able to worship openly in public, not being able to build new churches or synagogues, not attempting to convert any Muslim, etc.

In Spain, Muslims and Jews also had a long-standing working and cultural relationship with one another, beginning with the Umayyad conquering of Hispania in the 8th century. Under Visigothic Christian rule in Spain, the Jews were essentially slaves but rose up ahead of the Christians in the cultural pecking order after the Muslim conquering of Hispania due to much greater cultural similarities between Jews and Muslims than Spanish Christians. In al-Andalus (Muslim Spain), Jews would sometimes reach extremely high ranking positions in the caliphate, such as Samuel Nagid who was the highest ranking vizier for 30 years, starting at the beginning of the 11th century. Spanish Jews of this time were very arabized due to the Andalusian culture of tolerance that existed in Muslim Spain, therefore Spanish Jews would have shared far greater cultural knowledge and similarities to Muslims than Jews of other regions.

This doesn't answer your question completely but I think it gives a little bit of info as to possible reasons.

I'll try to answer your question, sorry if it doesn't satisfy you. Jewish community had supported the early Ottoman conquests, giving a positive image of Jews in the Ottoman Empire. Additionally, Jewish migration to the Ottoman Empire has been going on from the times of Mehmed II. The Jews also brought financial, diplomatic, and intellectual knowledge to the Ottoman Empire which was highly valued

Some background. For some Jewish commmunities, the Ottoman conquests of the 14th century meant potential liberation from subjugation, persecution, and humiliation from Byzantine hands. Because of that, Jews contributed significantly to the Ottoman conquests. The Jews of Bursa actively helped Osman's son Orhan capture the city in 1324. As a reward, he established the Jewish quarter to assure their autonomy in religious and secular matters. Jews entering the Ottoman territory were allowed to practice whatever profession they wished, to engage in trade and commerce without restriction, and to own landed property and buildings in town and country alike, in return paying a percentage of their revenues to the state in traditional Islamic manner. Further Ottoman conquests in Gallipoli (1354), Ankara (1360) and Adrianople (1363) were also accomplished with support from the small and impoverished Jewish communities which lived there under Byzantine persecution. And just like in Bursa, the Ottomans invited Jewish refugees from Europe to settle in and repopulate the newly conquered cities, providing them with substantial tax and other concessions.

After the conquest of Constantinople, Mehmed II faced some difficulties in rebuilding the capital. It had already been despoiled and depopulated by the Latin crusaders at the start of the 13th century, and there were few people and little wealth left by the time the Ottomans arrived. In imitation of what had been done by his predecessors he undertook strenuous efforts to repopulate and rebuild the city as quickly as possible. He tried to invite back Christians who had fled and initiated forced migration (sürgün) which increased the city's population to around 114,248 people. But this still leave the city relatively empty compared with its heyday. Mehmed could migrate only so many people for his capital from the conquered lands without depopulating them as well. That's why, Mehmed from the start attempted to encourage the emigration of Jews from Europe. Just as the Jews of Europe faced persecutions, the Ottoman state actively encouraged them to come and live in the Ottoman Empire under the same conditions of tolerance. This policy was further followed by his successors Bayezid II with the Iberian Jews and as a result, in the 16th century Jewish population in the empire reached around 300,000, making it possibly the largest Jewish community at the time.

Aside from repopulating cities, Jewish immigrants provided the Ottomans financial experience and skills valued in the Ottoman Empire. Jews served as physicians, financiers and political and diplomatic advisers which was considered more loyal than Christians. Many Jewish physicians came from the medical schools at Salamanica and Lisbon introduced the latest advances made in medical treatment in the West and served as physicians for the royal family and wealthier Muslim families. Quite a number of Jewish bankers managed to bring most of their capital into the Ottoman Empire from Spain and Portugal. They went on to engage in profitable and far reaching financial enterprises, making major contributions to financing trade, foreign exchange, lending, tax farming, and financing tax farmers. For example, in 1477 a partnership between a Jewish, Muslim and Italian trader paid 20,400,000 akçe for the customs duties at the port of Istanbul. Jewish immigrants also provided a substantial workforce for the Ottoman industry, especially textile production. For example, the Sephardic Jews of Salonika, had been assigned the job of manufacturing the woollen cloth needed for janissary uniforms. The Jews of Istanbul were also active in silk dyeing and tanning, and in addition to that, Jews also practiced many other crafts from tailors, carpenters, bakers, tinsmiths, butchers, and glassmakers, further enriching the economic life of the empire. Many Jews were also as money changer (sarraf). At the beginning of the 16th century, almost all sarrafs of Istanbul were Jews

Aside from financial benefits, many Jews also served as diplomatic agents for the Ottomans. As non-Christians, the Jews were viewed as as neutrals by the Ottomans. For example, Don Joseph nasi, probably the most successful and influential Jews of the Ottoman Empire at the time acted amongst other things as a diplomatic agent for the Sultan. In 1562 he was the principal negotiator in the negotiations between the Ottoman Empire and Poland, and in 1569 mediated the agreement signed in October between SeIim II and the King of France, Charles IX.

Sources:

The Jews of the Ottoman Empire and the Turkish Republic by Stanford J Shaw

A History of the Jewish Community in Istanbul: The Formative Years, 1453-1566 by Minna Rozen