This question contains two misconceptions in it - first, that Anatolia was converted easily and quickly, and second, that it was the Ottomans that did it.
First, the conversion of Anatolia religiously to Islam and linguistically to Turkish overGreek and Armenian, was a very, very long and very very slow process, in which the most dramatic phase was relatively recent, with the population exchange between the Turkish and Greek governments in 1923 [EDIT: and the Armenian Genocide before it].
In 1071 the Battle of Manzikert opened Anatolia to settlement by largely nomadic Seljuk Turks, who for the better part of a century had been migrating out of Central Asia to establish the Seljuk empire. Once Seljuk elites had taken military-political control of the region, a social division arose: Turks dominated marginal hinterlands suitable for grazing, while Greeks and Armenians populated the cities, towns, and agricultural areas - and it was the Turkish element that was the ruling one. Also, much of the coastland was under effective Byzantine, Venetian, or otherwise Christian control - the plateau was the Turkish heartland.
Conversion to Islam proceeded slowly across these centuries, not quickly, encouraged by political opportunity, economic reasons, avoiding the poll tax reserved for non-Muslims, and so on. Politically things were still fluid: Byzantines, Crusaders, and Armenian states would conquer bits of territory, and the Rum Seljuk state would fight against borderland Turkish principalities and would usually ally with Byzantium. Greek churches would gradually lose population and grow bankrupt; the Turkish and Muslim population slowly grew in relative terms.
By the early 1300s, when the Ottomans emerged out of the patchwork of beyliks formed when the Mongols shattered the Seljuk-Byzantine order, Anatolia was still a heavily mixed Muslim-Orthodox region (in fact, the early Ottoman armies were of mixed ethnicity and religion, as were those of other beyliks). Throughout the Ottoman period it remained so. Conversion did happen, and was largely unidirectional, from Christianity to Islam, but the peninsula remained diverse.
THE book on the subject of the Islamization of Anatolia is Speros Vryonis, The Decline of Medieval Hellenism in Asia Minor, which has remained a classic for a few decades.
I assume you are referring as to why the Turkish mainland is so heavily ethnically Turkish.
The Turks have been a presence in Anatolia since the 1000s, the great victory of Manzikert over the Byzantines gave them access to the Anatolian interior. As such the place has had over a thousand years to 'become Turkish'.
However by the turn of the 1900s there was still a large Greek, Armenian and Kurdish presence all over the Anatolian peninsular. Despite rising nationalism they were all still living quite peacefully together, although the strain of Muslims exiled from areas of Russia and later, the Balkans were putting strain and some religious tensions on the population.
It was in the last years and dissolution of the Ottoman Empire that the area you are speaking of really suffered ethnic tensions, especially under a particularly dark era under 'the three pashas'.
Commonly the dates given for the Armenian Genocide are given as 1915 onwards and the Genocide of Greeks from 1918 onwards. In 1922 I believe, the states of Turkey and Greece participated in one of the largest population exchanges in history, (there many Turks in the north of Greece and Crete too).
Let's split this down into smaller parts - "cleansing" (for lack of a better word) and assimilation.
Part of the homogenization* process did involve the Armenian genocide and the population exchanges with Greece. (This does not necessarily mean that there was a program of homogenization - things are more complex than that).
Now the real achievement was the assimilation. There are three kinds of assimilation that happened to the populations of Anatolia:
These three worked together. Converting people to a language is easy when all government and business affairs were done in that language. Religion is trickier, but it had political and economic advantages to convert and the religion was not too dissimilar from Christianity.
The third took time, but the identity of "Turk" became something closer to "American" than it is to a race. At some point, there were two words used - Türkeli and Türk, the former being someone-from-Turkey and the other being a person-of-Turkish-ethnicity, but people that assimilated to Türkeli/Turklander lost their identity and became a Turk.
Very few identities remain in Turkey today, some of which are Kurds, Laz and Arabs, but all are "free" to call themselves Turkish if they chose - it is actually better for them, politically and socially, to drop their "ethnicities" and be Turkish like everyone else. The Laz areas are some of the most nationalistic (Turkish nationality) regions of Turkey.
Turkey is not ethnically homogenous by any stretch of the imagination - the Ottomans (who themselves were less and less "Turkish" with every generation) succeeded in assimilating the peoples of Anatolia with language, religion and an inclusive ethnicity/national identity. Those that could not be assimilated (Armenians, Greeks, Kurds) were dealt with in other ways.