Segwit moved some transaction data in a new Witness spot. How do old nodes which do not run SegWit agree with the ones that do? They are not aware of the Witness part, how/why do they accept the same transactions as valid without that data?

When a block containing native segwit transactions is propagated to pre-segwit nodes, the witness scripts that unlock the segwit inputs are not sent; pre-segwit nodes think that segwit inputs are "anyone can spend" utxos, and consequently they don't expect a witness script for transactions that spend them, because no script is needed to unlock an "anyone can spend" utxo.