To use the Lightning Network, users will have to send a Bitcoin transaction to open a channel to a hub, and will possibly later need to send another Bitcoin transaction to cash out the value on the Lightning channel. The higher that normal Bitcoin transaction fees are, the more costly these opening and closing transactions become.
Suppose Bitcoin fees are $1. I want to use Lightning to send some micro-transactions. I only need to send about $5 worth of micro-transactions, but if I put just $5 on the Lightning channel I'll pay $1 for the Bitcoin tx fee. So I'm paying $6 to send $5, a fee of 20%. Worse, if the hub I was using became uncooperative after I had spent $2.50 of my initial $5, I'd be forced to send another Bitcoin transaction to get my money back, so I'd end up having paid $2 in fees to send $2.50 in value -- that's 80% in fees.
To get around the high $1 Bitcoin tx fee, I'd probably want to put larger amounts of money on the channel at a time. If I sent $100 to the channel and I spent it all so I didn't have to close the channel, I'd only be paying 1% in fees. However, if I'm poor it may be inconvenient for me to lock up such large amounts of money on a Lightning channel. Also, the hub could still become uncooperative and force me to pay another Bitcoin tx fee to get my money back. If the hub did this after I'd only paid $2.50 again, then I still would have effectively paid 80% in fees to send $2.50, even though I tried to plan ahead and make a large deposit to keep my amortized fees low.
Even with the Lightning Network, high Bitcoin fees increase the amount of money users need to tie up to avoid paying high amortized fees, and they allow users to be harmed more when their counterparties don't cooperate.