Your internet connection is delivered much like this--
(Internet) -- [Backbone provider] --interconnect-- [ISP] --last mile-- [you]
the Last Mile problem is how to get the service from an ISP's central office or local point of presence to the actual subsciber's house. This is where cable has the problem. Cable is a fiber-to-the-node type system, the cable company runs fiber from their central office or PoP to a node located in your neighborhood, which then sends the signal out over the coax line. This coax is then delivered to many homes, which all share its bandwidth. If you put too many homes on one coax line, that bandwidth will be fully saturated and everybody's connection will slow down.
Verizon FiOS uses fiber to the home. While it's a similar system, in that the fiber is optically split for each subscriber and many subscribers share one fiber (much like cable), the resulting bandwidth is so much greater that this isn't considered to be a problem.
However these are both last mile problems. What's going on with FiOS is an interconnect problem, between Verizon and their backbone provider. All the FiOS customers want Internet data, and even if it can zip around Verizon's network really fast, they want to get out of Verizon and get things from the greater Internet. So Verizon pays a larger backbone provider (in this case Level3) for this access. This is how it always works- the ISP pays a backbone for bandwidth, which they resell to their subscribers via their own (ISP's) network.
Verizon is saying that because Netflix's traffic comes in over their Level3 backbone interconnect, Level3 should PAY THEM for the privilege of serving FiOS customers. This is not how things work and not how things have ever worked.
To make an analogy- let's say Level3 is a cattle farmer, and Verizon is McDonalds. Right now, McDonalds pays the farmer for beef which it then marks up and sells to the customer in the form of a cheeseburger. Now imagine that McDonalds tells the farmer that THE FARMER NEEDS TO PAY McDONALDS for 'access' to the hungry customer. Ass backwards, right? That's what Verizon is trying to pull right now.
Since Level3 refuses to pay, Verizon refuses to upgrade their interconnect links. So all the FiOS customers (who aren't using much of their last mile links) are fully saturating the link between Verizon and Level3. L3 has begged Verizon to upgrade these interconnect links, but Verizon refuses to make the upgrade, because they say L3 should be paying them instead of the other way around.
Now it's useful to understand- there ARE situations when payments like this take place- when two major backbone providers connect with each other. So if L3 and say Global Crossing, or XO, or one of them set up a peering arrangement between each other, and more traffic flowed in one direction than the other, the company sending more traffic would generally pay the company receiving traffic. These companies don't service customers directly, they sell access to ISPs)
This doesn't apply to Verizon though, because Verizon is not a backbone network in this context. They are a consumer-level ISP, who must pay for backbone access to resell to their customers. Verizon is trying to play both sides and get paid on both ends, which is NOT how things are done.