As this seems not to have been answered:

Litecoin wants Segwit because it allows Atomic cross-chain trading with a simple mechanism very similar to the one that would be used in the Lightning Network (see: https://en.bitcoin.it/wiki/Atomic_cross-chain_trading). Atomic cross-chain trading would be a major advantage for all altcoins, because there would be a simple and trustless way to change value from one blockchain to another. So to buy altcoins with BTC you wouldn't have to trust an altcoin exchange (remember Cryptsy?).

Obviously they want it also to make it appear that they are technically on the forefront of the cryptocurrency movement and have an excuse for a pump.