When a hardware wallet signs a transaction, it can easily verify the total amount being spent, but can only safely determine the fee by having a full copy of all the input transactions being spent, and must hash each of those to ensure it is not being fed false data. Since individual transactions can be up to 1MB in size, this is not necessarily a cheap operation, even if the transaction being signed is itself quite small.

Segwit resolves this by explicitly hashing the input value. This means that a hardware wallet can simply be given the transaction hash, index, and value (and told what public key was used), and can safely sign the spending transaction, no matter how large or complicated the transaction being spent was.
Segwit improves the situation here by making signature data, which does not impact the UTXO set size, cost 75% less than data that does impact the UTXO set size. This is expected to encourage users to favour the use of transactions that minimise impact on the UTXO set in order to minimise fees, and to encourage developers to design smart contracts and new features in a way that will also minimise the impact on the UTXO set,Because segwit is a soft-forking change and does not increase the base blocksize, the worst case growth rate of the UTXO set stays the same.