What makes running so much less energy-efficient than bicycling?
One word: inertia. When you're riding a bike on a level gradient you just need to give it a push to get going, then you can coast for quite a while before friction and air resistance slow you down. In other words, the relatively frictionless wheels mean the bicycle's kinetic energy doesn't dissipate quickly. But the human body doesn't have wheels, so while running you have to give a good kick to get going, and then another kick to keep going on the next step, and so on. When hills are involved the difference is even more pronounced, since we run downhill the same way we do on the level, by continually pushing ourselves forward; whereas on a bicycle you can take advantage of the slope and just coast down it.
I suspect that raising and lowering your centre of mass isn't as inefficient as the other answers have suggested. This is because your legs are springy, so at least to some extent you're just converting energy back and forth between gravitational potential and the spring force in your legs. Humans are possibly the most efficient long-distance runners in the animal kingdom. There is a school of thought that says the reason we are bipeds is that we evolved as endurance hunters, chasing our prey until it collapsed from exhaustion rather than trying to outrun it over short distances. Whether that's true or not, we probably wouldn't do all that bouncing up and down if there wasn't a good reason for it.
You might ask why, if using wheels is so much more efficient, didn't we evolve that instead? I don't know, but it seems no animal has been able to evolve wheeled locomotion.
Many of us have ridden bicycles at some time in our lives. and in fact this mode of transportation has become markedly more popular recently as a result of the energy shortage. Each morning at my own university, Duke, people can be seen riding machines with masses of $10$ to $20$ kilograms and struggling to reach one of the campus entrances at the top of a long, steep hill. As in many other aspects of animal locomotion, there is a paradox here. Why should people encumber themselves with such heavy apparatus, particularly while going uphill? Ask a rider this question, and the response is usually: "It's easier than walking" or "It's faster than walking." But why should it be?
A number of incorrect explanations are offered: "A bicycle has gears." Shifting gears allows the rider to vary the speed at which the feet move; but even if the foot speeds of a cyclist and a pedestrian are matched, the cyclist still goes farther and in less time on a given amount of energy than the pedestrian. "Your weight is supported by the seat." But if you pedal standing up, biking still is faster and less costly of energy than going on foot. "Your center of gravity doesn't go up and down." But it does if you pedal standing up. Why, then, is bicycling easier than walking or running?
[…]*
We can now appreciate why bicycle riders are willing to propel the extra weight of a bicycle, even when going uphill. The cost of transport on a bicycle is low because active muscles are not stretched while pedaling, and mean muscle efficiency is about $.25$, nearly its maximum value. The wheels stabilize the rider's center of mass. Even if the rider accelerates the center of mass vertically by pedaling while standing up, active muscles need not be stretched. When the center of mass falls, the cranks, sprockets, chain, and rear wheel constitute a system of levers that transposes the vertical motion to a horizontal one by supplying a perpendicular force. Thus, humans can use external machinery to move along a level surface with the same muscular efficiencies that swimming and flying animals achieve naturally.
The Energetic Cost of Moving About: Walking and running are extremely inefficient forms of locomotion. Much greater efficiency is achieved by birds, fish—and bicyclists. V. A. Tucker, American Scientist, Vol. 63, No. 4 (July-August 1975), pp. 413-419
*Of course, most of the article is where I put "[...]". It's quite a good, fun read. There's even some kind of Galilean experiment with dropping pigeons and rats from heights.
Bicycles make better use of inertia/momentum. As Nathaniel said, one push and you can coast for quite a while. That's just not possible while running.
Running wastes energy moving up and down. In addition to moving forward, running requires a substantial upward push to get your body airborne, giving you time to bring your other foot forward. You then cushion and spring forward and upward again. While bicycling does have an up-and-down component to the pedaling, because the bike doesn't leave the ground, the energy you use in pedaling is converted much more efficiently to forward motion.
Bicycling can translate weight to propulsion. While most serious bicyclists will tell you that pedaling is about spinning, not stepping, any 10-year-old can tell you that going butt-in-the-air and transferring your weight from left to right gets you going pretty quickly.
Pedaling with toe clips makes use of the entire leg motion. When your feet are locked to the pedals, it isn't just the pushing-down portion of the pedal stroke that is used; lifting your foot, pulling it forward, pushing down and pushing backward all keep tension on that chain and so add power to the stroke. When running, fully half of your foot's cycle is wasted energy from a forward-motion perspective.
Bicycling gives you mechanical advantage. Even with a single-gear bike, the motion of your foot is magnified when translated to the wheel. On a multi-speed bike, the ratio of the top gear is pretty high indeed. This allows two things; first, your effort is magnified, and second, your tempo slows, which reduces the amount of energy wasted moving the weight of your legs around.