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Lightning Network still requires broadcasting transactions to the network when you want to open a channel or close and old onel. That's why some people say that we will have to increase the blocksize anyway. Lightning Network is still not widely used among merchants so I don't think that is the reason of low transaction fees. It looks like batching and growing SegWit adoption does the good job of off-loading the load.
we need to increase the capacity and increasing the blocksize is not the only way of doing that. for starters SegWit is increasing the capacity. there are also other solutions such as signature aggregation and schnorr,... that can increase the capacity in the future. we may eventually increase the blocksize also but it seems to me that is a long ways down the line.
as for LN, from what i see, the way it can help scaling is that it has the potential to move a large number of transactions out of blockchain. for example imagine all the major exchanges (coinbase, kraken, bitfinex, bitstamp, poloniex, binance) started using LN, blocks may become mostly empty in that case and you don't need to open and close channels each time you make a transaction.