So. Once more, with feeling: Here are the reasons bicyclists should not be required to pay a special license fee.
1) Cyclists already pay taxes that pay for roads. Most cyclists already have drivers' licenses (because most cyclists also drive cars); in addition, roads are paid for with taxes that are paid for by everyone, including sales taxes (which you pay every time you make a purchase) and property taxes and levies (which you pay when you pay your mortgage or rent). The Victoria Transport Policy Institute has a good explanation of how cyclists and other non-drivers subsidize roads for automobiles here.
2) The externalities associated with riding a bicycle are positive, whereas the externalities associated with driving a car are negative. This is relevant when we're talking about who should pay for which mode of transport: Drivers pay, in part, because driving has negative consequences for society in general.
Specifically: Driving increases a society's overall carbon emissions, promotes suburban sprawl (and the destruction of our dwindling rural and farmland areas), and increases the amount of our urban areas that is given over to highways (e.g., the Alaskan Way Viaduct) instead of, say, city parks.
In contrast: As more people ride bikes instead of driving, positive externalities are produced, including: Cleaner air (because bikes, unlike cars, don't produce carbon emissions), healthier people, reduced traffic congestion, a lower rate of obesity, lower health-care costs, and so on.
3) The reason states require drivers' licenses is that driving a car is inherently dangerous to other people, and because cars are far more likely to produce (expensive) fatalities than bikes. (This, by the way, is also the reason states require drivers to be a certain age, typically 16, while children learn to ride bikes as young as 4 or 5.)
In other words: Drivers' licenses are tied to the use of cars, not the use of roads. In the same way that doctors are required to have a license to perform surgery (on the premise that surgery is dangerous and difficult), driving is a dangerous and difficult activity; therefore, people who want to drive should be qualified to do so. (This is why, for example, we have several "classes" of driver licenses; the more difficult a "class" is—e.g., driving an 18-wheeler big rig—the more hurdles there are to obtaining the license). Cycling is neither (inherently) dangerous nor particularly difficult; therefore, like walking, we allow people to do it for free.
In contrast, one of the main reasons for charging drivers to obtain a license is that automobile operators are responsible for tends of thousands of fatal accidents every year. License fees offset some of this expense to society. Requiring cyclists to pay a license fee would be like requiring pedestrians to pay a toll to use sidewalks—an absurdity that most members of society (including, I think, drivers) would agree disincentivizes behavior (cycling) that's beneficial to everyone.
4) Bicycles have a minimal impact on streets; cars (and buses, and trucks) have a maximal impact. Bikes don't significantly damage city streets, whereas cars, trucks, and buses do. So those who argue that bicyclists should pay "their share" for street maintenance aren't actually considering what cyclists' "share" is, compared to the cars, trucks, and buses that are producing the expensive potholes and other repair problems the city must address.
5) Most cyclists already have driver licenses. Therefore—by the faulty standard set by those who say cyclists should "pay their share" too—they're already paying for the roads. So those who say cyclists don't pay their share are simply wrong, even by the faulty standard that says cyclists should pay as much as drivers, despite the fact that cycling is a net benefit to society.