I know that the brakes are to prevent them from flying apart. But why can't they spin at the same revolutions but produce more electricity, act as a bigger generator in a stronger wind?

Most commercial turbines operate in a range from 8 mph to 25 mph. They cannot handle the increased tork it's a tradeoff for being able to generate at lower speeds. Smaller units can take far greater wind speeds but have trouble generating much electricity at speeds under 10 mph. Two months ago in a northern N.Y. wind farm one turbine was damaged and another was destroyed when they failed to shutdown the winds were gusting to 25mph.

This is to keep the system operating in the "sweet spot" of the unit. This is the part that the unit will generate the most electricty. This is done because every place has a average wind speed. If you are producing electricty in the range where the wind is blowing the most (avg) you will get the most electrical production from the unit. It is also for safety, keep in mind these units are spinning due to the wind, if the wind is more than the unit can handle it will spin out of control and begin to break apart.

Anthony Stonis
Building Energy Experts
www.BuildingEnergyExperts.com

I understand the turbines will self-destruct if they spin too fast. But isn't there a way to increase the resistance and load (to produce more energy) when the wind blows harder, thus keeping the same RPMs as when the wind is less?
I am having trouble explaining this. Have a turbine that performs like one with a bigger generator during high wind situations so as not to waste that wind power by braking.

The amount of mechanical load "seen" by the turbine that regulates it's speed is controlled by modulating current in some of the generator's windings.  Real-world wire is not super-conducting, and it gets hotter with higher & higher currents. Getting the heat out of the windings is one of many critical engineering tasks for durable designs.  In order to control heat at higher current the windings have to be fatter- at some point there's just not much you can do to keep it from thermal runaway, since the windings can only be so fat before the size & mass of the thing balloons to ridiculous proportions, and the low/moderate speed efficiencies suffer with higher mass generators. Getting the mass out has also been a critical design feature.  All real-world designs are a compromise.

American Superconductor has done very well with the explosion of wind turbine production, since they can pound several times the amount of current through their (very fancy, not-so-cheap) wire compared to standard material of equal volume or weight.  They've gotten into designing power factor correction sub-systems for grid-connecting wind turbines using their own material as well. See:  http://www.amsc.com/products/applications/windEnergy/index.html

But company name notwithstanding, the conductivity of their wire has limits as well.  At some wind speed you have to also throttle it back aerodynamically to keep the windings from cooking.

Instead of brakes, what about constantly varying the gearing to achieve the same thing?

This can still only help the situation so much but there are turbines that employ regenerative braking meaning that when the brakes are used at least you're gaining some electricity back from them - I'm not sure whether it the electricity gain is coming from the heat or transfer of kinetic energy or both.