Why the Indo-European language didn't spread to Central Asia?

Indo-European languages did spread to Central Asia, and are still spoken in Central Asia. Not counting the relatively modern spread of Russian into Central Asia, surviving and thriving Central Asian Indo-European languages include Farsi (Persian), Pashto, and Tajik among others.

If you restrict "Central Asia" to the former Soviet republics (thus excluding Iran and Afghanistan), this restricts the major Indo-European languages to Tajik, which is the official language of Tajikistan and a recognised minority language in Kazakhstan, Kyrgyzstan, and Uzbekistan.

Indo-European languages dominated Central Asia in the past. Notably, Sodgian was the most important Indo-European language in what is now "the stans", and was the lingua france for trade with both east and west of Central Asia along the Silk Roads. Written material survives from not only former Sogdia, but also Xinjiang. Other major Central Asian Indo-European languages of the time include Bactrian and Khwarezmian/Chorasmian. Early important Indo-European languages of the area include Scythian (which has left Ossetian as a surviving relative, via Alanic).

What happened to these languages in Kazakhstan, Kyrgyzstan, Turkmenistan, and Uzbekistan? Answer: the Turkic expansion resulted in large Turkic populations speaking Turkic languages in much of Central Asia by the 10th century. Turkic and Indo-European languages existed side-by-side, with both being major languages (and they still exist side-by-side, even in Turkmenistan (about 5% old Indo-European languages)). However, in the stans other than Tajikistan, Turkic languages became the dominant languages over time. This took a long time. Even though Turkic people and Turkic political power were important already in the 6th century, Indo-European languages remained dominant in literature and government for a long time. For example, the official language of the Turkic Khaganate (Göktürk Khaganate) of the 6th to 7th century appears to have been Sogdian. Even after Turkic languages became spoken more widely, Indo-European languages (notably, Persian) remained important in government and as literary languages for many centuries - one of the first major Central Asian literary works in a Turkic language is the 16th century Baburnama (the memoirs of Babur, the founder of the Mughal Empire), written in Chagatai Turkic. Even then, Persian was so important as a literary language that the Baburnama was translated in Persian later in the 16th century.

And an interesting extra: the relationship of the Chinese languages to other languages is uncertain. They are, mostly, classified with Tibeto-Burman languages as Sino-Tibetan, but the differences between Chinese languages and Tibeto-Burman languages are large, and this grouping remains uncertain (see Zev Handel, Zev, "What Is Sino-Tibetan? Snapshot of a Field and a Language Family in Flux", Language and Linguistics Compass 2.3, 422-441, (2008) and George van Driem, "Sino-Austronesian vs. Sino-Caucasian, Sino-Bodic vs. Sino-Tibetan, and Tibeto-Burman as default theory", pp. 285-338 in Yogendra Prasada Yadava et al. (eds), Contemporary issues in Nepalese linguistics, Linguistic Society of Nepal, Kathmandu, 2005). Beckwith (in Christopher I. Beckwith, Empires of the Silk Road: A History of Central Eurasia from the Bronze Age to the Present, Princeton University Press, 2011) suggests that Chinese languages might be Indo-European creoles (or possibly Indo-European languages).