How does Segwit reduce transaction fees?

[squatter]

As far I know a fork does not have any impact on the orginal blockchain.

That depends on what kind of fork it is. For example, a soft fork that occurs without majority miner enforcement can cause a “wipeout attack” on the original chain. Replay protection is not possible in soft forks, nor would it be required. So, this can happen in a UASF:

1) UASF occurs with very little hash support.
2) The majority of transaction volume moves to the UASF fork, even though it’s the minority fork.
3) Miners, driven by profit motive, switch to the UASF fork.
4) Eventually, the UASF chain gets more cumulative work than the legacy chain. This causes a chain reorganization that “wipes out” all transactions on the legacy chain going back to the fork block.

That means that anyone who received a payment between the UASF block and the reorg will lose that money. Given the time involved, it’s possible that thousands of confirmations could be undone. This is an extreme scenario, but it’s good to be aware of.

[Aura]

Transactions need less space on the blockchain when SegWit is used. That means miners can take more transactions in a block, so compared to legacy transactions it's much more efficient. The result is lower fees than if you would use legacy transactions, because the fee is shared over more transactions than without SegWit.