Until now, this hasn't been a problem, because most people didn't max out their connections, and if they did they didn't do it for very long. The very nature of Netflix's business, though, is to serve the best traffic that the customer's connection can handle, and that starts to add up. Unfortunately, it adds up so much that in some cases (e.g. Comcast) you can't even serve VHS-quality content to all the customers who want it.
Because Verizon is overselling their bandwidth, they're now having issues where they can't service the demand that they sold their customers on. Instead of upgrading their upstream connections (which would cost them money, which they have a lot of), they want Netflix to pay for it. The logic is that Verizon wouldn't have these problems if Netflix weren't sending so much content; that argument breaks down when you consider that Verizon sold their services based on available bandwidth and now can't provide it; Netflix is just the service that exposed the problem.
Netflix has other solutions, though; they can provide servers that get hosted in the ISP's network to provide content to users without traffic coming from outside the ISPs' networks (thus freeing up their normal uplinks for regular traffic), or by peering with Netflix at various points (which accomplishes the same thing).
Unfortunately, neither of these solutions involves Netflix giving money to Verizon, which is why Verizon doesn't want to bother with going to the time and trouble when they can force Netflix to pay for their infrastructure upgrades instead.