From r/AskHistorians:
https://www.reddit.com/r/AskHistorians/comments/1brxju/from_where_did_indo_european_come/
https://www.reddit.com/r/AskHistorians/comments/13m0pm/europe_before_the_indoeuropeans/.compact
More details:
http://www.archaeology.org/exclusives/articles/1302-proto-indo-european-schleichers-fable
I'm not a historian, but I am a linguist who has studied historical linguistics a bit, so I will provide at least a sketch of an answer: the basic answer is human migration. In the case of the Indo-European family, the basic idea is that thousands of years ago, people who spoke Indo-European languages migrated into south and central Asia and into Europe.
Later waves of migration (e.g., the ancestors of today's Turks and Hungarians) occupied some of the land between the European IE-speakers and the Asian IE-speakers.
These geographically intervening populations reinforce our naive conception that the languages of Iran and northern India and those of western Europe must be very linguistically distinct, when in fact Spanish and Hindi are a hell of a lot closer to each other than either one is to, say, Turkish.
It's not that rare for a language family to range through a wide portion of the globe. Austronesian languages are spoken from Hawaii and Easter Island/Rapa Nui to Madagascar. Niger-Congo from Senegal to South Africa. Afro-Asiatic from Iraq to Mauritania and Somalia.
A lot of migration and assimilation can happen in thousands of years, and Proto-Indo-European predates recorded history.