How were the Vedas memorised word for word for thousands of years?

They were memorised with a system of cross-checking recitations, not just by repeating the text. Vedic chant is often considered the oldest unbroken oral tradition in existence, and the recitation styles, called pathas, are designed to allow the complete and perfect memorisation of the text and its pronunciation, including its pitch accent. Eleven such ways of reciting the Vedas were designed.

Here is the trick. Students are first taught simpler methods, such as continuous recitation and word-by-word recitation, with a conscious pause after every word. Then the pairs: a hymn "word1 word2 word3 word4" is recited as "word1word2 word2word3 word3word4". The advanced modes go further. In the ghana-patha, the "bell" mode, the words are repeated back and forth in a bell shape. The eight complex modes involve reversing the word order, and in Vedic Sanskrit chanting words backward does not alter their meaning.

Why go to such trouble? Because the sound itself mattered. The insistence on exact pronunciation and accent is tied to the belief that the potency of the mantras lies in their sound. In the transmission of the Samhitas the emphasis is on the sound of the mantras, not their meaning. So the goal was not to remember the gist but to preserve the sounds exactly.

And the results were checkable. Each text was recited in several ways, so the methods acted as a cross-check on one another. All 1,028 hymns, with 10,600 verses, of the Rigveda were preserved this way.

There was also a person in the loop. Only a living teacher, tradition says, can teach the correct pronunciation and explain hidden meanings, in a way the "dead and entombed manuscript" cannot.