Without getting into my misgivings about the studies, the general idea is that the ethnic Turks, by virtue of being the elite of society, managed to assimilate the "natives" of Anatolia. Which, frankly, doesn't make a lot of sense. As late as 1906 large swathes of the Turkish coastline were still ethnically Greek (the map was published in 1911 using data from 1906, hence the label). The top-down assimilation model doesn't really do a good job of explaining this. Especially since, at that time, the Ottoman Empire had held Constantinople for 453 years and other places for even longer. Adana had been held for at least 835 years and yet the map still suggests a sizable Armenian demographic.

All of this is compounded by the fact that the Ottoman Empire was remarkably tolerant of ethnic minorities. With the sole exception of Devsirme, it didn't care at all what your personal faith or language was as long as you kept the peace. If the empire was so utterly indifferent to your faith (even while your sultan claimed the title Caliph and held the three most holy cities of Islam), why would it make any effort at all to convince you to become Turkish? In fact, the Ottoman Empire typically made sparing use of terms like "Turkish" because it wanted to avoid diminishing its non-Turkish population (you can see this as early as Mehmet II's reign, where his personal addresses to his soldiers at Constantinople in 1453 avoid explicit mentions of Turkishness).

If you're interested in an incredibly dense read that illustrates the general equality of Ottoman law, I recommend Michael Ursinus' Grievance Administration (Sikayet) in an Ottoman Province (specifically Rumelia). It's basically a summary of a whole mess of court cases that went before an Ottoman beylerbeylik (provincial governor) between 1781 and 1783. And, with the sole exception of noting who were the Muslims and non-Muslims, it appears to be remarkably even-handed.

I'd describe the "assimilation" of Anatolia as working both ways. The Turkish elite certainly did borrow customs and vocabulary from non-Turks. Ottoman Turkish itself draws something like 80% of its vocabulary from Arabic or Farsi (and the Ottoman Turkish alphabet is basically Arabic script with a few added letters). The Turks most certainly did not retain traditions of their Central Asian ancestors for terribly long; the timariot system looks more like western knighthood than the tribal systems that characterize Central Asia of the time. As the new overlords of Anatolia settled, and became more like their subjects, so too did those subjects become more like their overlords.

It's also important to remember that Anatolia is geographically situated between Europe and Asia (assuming you don't feel like diverting way north) and is effectively the meeting point of those two continents. Genetic diversity of the sort we see emerging from these studies is pretty much exactly what we should have expected. You don't have an area situated at the crossroads of continents without extensive cultural exchange.

The answer seems to be contained in your question: cultural assimilation. It's been common too in the formation of Europe's nation-states, whose frontiers rarely coincide with past (and inevitably fuzzy) ethnic or linguistic boundaries. England is a prime example, with Anglo-Saxon arrivals thought to have been only a minority in a predominantly Celtic-language area that soon adopted a form of their Germanic tongue. The precise mechanisms are unclear: higher status as a conquering people; socio-economic privilege and demotion of the native group to second-class status, sometimes with fewer rights; possible physical relocation of the indigenous population from the best lands, during or after conquest. But it does seem that thorough transformation can be effected in a matter of centuries provided that the newcomers are sufficient in numbers (for instance the Normans never imposed their language despite their radical social and political impact).