Pedaling at different gear ratios on flat terrain: what's the point? I have seen many cyclist pedaling at different gear ratios on the same (flat) terrain and cruising at (roughly) the same speed. They move quite fast, but when I look at how fast they pedal, I realize that the gear ratio is much lower than what I'm used to set for same speed and same inclination. To attain the same speed, I must use a higher gear ratio, but somehow they manage to use a lower one by pedaling very fast.

Is there an advantage in cycling this way? How do the different training gear ranges affect my fitness in the aspect of power vs. speed? (Power - fitness in climbing uphill, Speed - highest speed at no inclination). I have attempted this technique, but I find it quite difficult to balance and have a good grip of the bike.
First, different people have different optimum cadence.  So that's one thing that explains different cadence between different cyclists.

Second, to travel at any given speed requires a certain amount of power.  You can get more power by pushing harder at the same cadence, or you can pedal faster at the same level of force (or you can pedal harder and faster..).  

Ever lift weights?  Pick a weight you can do ten times, but no more.  You can't lift ten times that much once, can you?  You get much better endurance at lower levels of force.  It's the same with pedaling a bicycle - for a given power level, most cyclists get better endurance by pedaling faster at lower levels of force, at least up to a point where your muscles start getting less efficient because they can't contract fast enough.  There's actually a good bit of research on this, and there are physiological reasons why it's true.

But in general, if you want to go fast for a long period of time, it's better to pedal easy and fast than it is to pedal hard and slow.

It's tough to get used to at first, but after a while it will feel a lot more natural.  Just get something that measures cadence and you can start working on riding at higher cadences.