It was an evolution, a slow growth. I don’t think the Vedas had the effect that the Ten Commandments had on the Hebrews or the Quran on what are now Muslims. It wasn’t “here are the Vedas... boom you have a religion to fit into”.
The Vedas were complied over centuries by many different rishis (sages) in a pastoral and illiterate society. For centuries they were passed down orally. Much of the Vedas is dedicated to gods of nature... Agni, Vayu, Dyaus Pitr, Varuna, Prithvi, respectively: fire, wind/air, the sky, waters, the earth. There is much mention of horses, which were extremely important to most Indo-European peoples.
Word and teachings spread. Keep in mind that the Vedas are apaurusheya, “not of man”. That is, they were divinely inspired to the rishis. There was no prophet, messenger or voice from the sky. Hence their gradual growth and spread.