The witness data is not separated from the transactions, it's in each transaction before the nLocktime field.   It prevents malleability attacks by separating it from the txid so that child transactions aren't invalidated by changes to the signature, the same way *within* a transaction the various signatures don't sign each other.  It doesn't change the space of anything.  It doesn't directly lower fees, but because the witness field isn't counted in the original blocksize limit the change was able to effectively eliminate the old limit and replace it with a "weight limit"-- like measuring blocks by the pound instead of by the gallon and the witness data takes up less weight.  Nothing in bitcoin sets the fees for transactions except market forces, but the increase in capacity can result in lower fees for a given amount of demand since users bid against each other for access to capacity.