A lot of research went into trying to define language families. It is a very interesting topic to read about. https://en.m.wikipedia.org/wiki/Proto-Indo-European_language

The TL;DR version is that languages evolve in predictable ways. You can reverse engineer what a parent language would sound like by comparing its daughter languages. This is very similar to cladistic analysis in evolutionary biology. By comparing traits of organisms it is possible to reverse engineer how they evolved.

Like most scientific hypotheses the evidence for language families was pretty flimsy at first, but it had accumulated over time and is supported by archeological finds, and DNA analysis. All languages should have a common ancestor language far enough back, in the same way all organisms have a common ancestor. Unlike organisms, language does not have fossils and the language groups shown here are basically as far back as we are able to show similarities to languages.

India is one of the most interesting countries in the world linguistically because it has multiple language families. Because children tend to speak their parent’s language, the language families correspond to human migrations. The Dravidian speakers of South India used to be more wide-spread in India before proto-indo-europeans orginating from central asia displaced them. This is an incredibly interesting topic to read about. https://en.m.wikipedia.org/wiki/Peopling_of_India