Proto-Indo-Europeans speakers lived on Anatolian Homeland
In order to decipher where the Proto-Indo-European (PIE) speakers lived, many scholars have first hypothesized what this language looked like. In order to accomplish this, linguists use a process called reconstruction, which can allow them to establish the vocabulary and grammar of unattested--or undocumented--languages. One method of reconstructing is known as the comparative method, which is used to illuminate the features of a common ancestor language (or protolanguage) using two or more of its descendant languages. In the comparative method, linguists use lists of cognates, which are words in several descendant languages that come from the same ancestor language; English "hound" and German "hund" are cognates, for example. Using these cognate lists along with research on how sounds in language typically change over time, linguists then attempt to posit the appropriate form of the word in the ancestor language. However, linguists may be unable to produce an ancestral form if they do not have enough evidence; there may not be enough cognate words documented, for example. Therefore, the comparative method does not always work, but it is useful in helping scientists theorize about what an ancient language may have looked and sounded like. The comparative method is essential for locating the Proto-Indo-European homeland because reconstruction allows scientists to understand the vocabulary of the PIE language. For example, if an ancient language has many words referring to sand and dry ground but no word for 'snow,' it is safe to assume that the speakers of this language lived in a hot, desert-type environment rather than in a forest or on a mountain. Additionally, if an ancient language has words for specific types of animals or plants, linguists can work with ancient zoologists and botanists to triangulate where its speakers may have lived based on where those animals and plants were located long ago.
Speakers of pre-Proto-Indo-European, or at least an early or archaic form of the language, originally lived in ancient Anatolia around 7000 BCE and spread from there. Using the process of linguistic reconstruction, some researchers have concluded that the now-extinct Anatolian branch of languages (including Hittite, Luwian, and others) was the first to branch off from Proto-Indo-European (PIE). Linguists point to features present in Anatolian languages that are missing from other descendant languages in the PIE family as well as the reverse--there are technological vocabulary items, for example, missing from Anatolian languages, such as the words for 'wheel' and 'wagon,' that suggest that the Anatolian family existed prior to the creation of these technological innovations. Archaeological evidence also shows that neolithic Anatolia was inhabited by early farming communities that subsequently spread into the Balkans and beyond. This must have been how PIE speakers began their spread into Europe and Asia.
If the Anatolian family broke off so early in the history of PIE, the former would be vastly different than any PIE language by the time Anatolian languages are written down several millenia later; however, this is clearly not the case: languages like Hittite are still obviously similar in many ways to other PIE languages by this time. The archaeological evidence demonstrating that farming communities existed in ancient Anatolia does not automatically point to pre-PIE either, and many scientists believe that these neolithic residents were actually speakers of a non-Indo-European language and therefore completely unrelated to PIE.
Rejecting the premises
More on this hypothesis: https://www.proto-indo-european.ru/ie-cradle/_pdf/clouds-over-ie-homelands-nallory.pdf