Why do Sanskrit, Greek and Latin share so many words?

They share words because they are relatives. Sanskrit, Greek and Latin all descend from one much older language, which linguists call Proto-Indo-European. In 1786 Sir William Jones lectured on the striking similarities among the three: he argued that they bear a stronger affinity, in the roots of verbs and the forms of grammar, than could have been produced by accident, so that no philologer could examine them without believing them to have sprung from some common source. Sanskrit belongs to the Indo-Aryan branch of that Indo-European family.

The clue had been noticed long before. In 1585 a Florentine merchant named Filippo Sassetti, travelling in India, noted word similarities between Sanskrit and Italian, such as sapta and sette for seven, and nava and nove for nine, but his observations led to no further scholarly inquiry. And although Jones is credited with discovering the family, the first scholar to suggest it was a Delhi-based Mughal scholar, Siraj-ud-Din Ali Khan Arzu, in a Persian-language treatise.

The family is huge: over 3.4 billion people, 42% of the world, speak an Indo-European language as a first language. Sanskrit matters to it because of the Rigveda, whose hymns are notably similar to the most archaic poems of the Iranian and Greek language families. Because the Rigveda was orally transmitted with exceptional rigour and fidelity, it is of vital importance in reconstructing Proto-Indo-European.

How a relative of Latin came to India is still argued over. The academic consensus places the family's homeland on the Pontic–Caspian steppe, and the Indo-Aryan migrations theory has the speakers of what became Sanskrit arriving from the north-west in the early 2nd millennium BCE. Aryan indigenist writers disagree, holding that the family originated in India.