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During all Star Treks, if a system isn't performing the way they need, they always reroute power from other systems.
How does this power surge cause systems to work better?
I always thought that if an electrical system got more power than what was designed, it would overload, but not 'work better'.
It comes down to how the individual systems work and what the goal of adding more power is.
In some cases, such as with life support as @jwenting mentioned, the extra power may simply be to ensure continued operation of the system if primary and standard secondary sources of power are interrupted.
With weapons the extra power would be to increase their destructive power or place them in an overloaded state, as @apoorv020 said.
Extra power to shields is used to maintain the shields when the standard capacitance of the shield generators is reduced to 0 and to restore their full power more quickly when not under fire.
With engines the effect is similar to weapons, but with the goal of increasing their output for more speed (think of reving the engine in your car in to the red zone).
In most cases routing additional power to a system for extended periods is probably not a recommended action. I would also bet that after the episode the engineers probably pull a couple extra shifts to check for and repair any damage caused by minor overloads. But in a pinch routing additional power to one system or another can turn the tide in a battle, get you somewhere a little bit sooner or increase the "grip" of your tractor beam to move that large asteroid.
There's another possible explanation: In your question, you said: "I always thought that if an electrical system got more power than what was designed, it would overload, but not 'work better'."
However, that only applies to the type of circuitry used nowadays in reallife. At the time of The Original Series, transistors/resistors/etc had been replaced by duotronics. By the time of The Next Generation, duotronics had been replaced by isolinear circuitry. They may well react entirely differently to surges in power, than our modern circuits.
It actually makes some amount of sense that routing more power to system would make them work better - it's possible that there's a limited amount of power on board a starship, but that the equipment is designed to accommodate much more in the event of an emergency.
However, one of the things I always found a bit implausible in Star Trek was when they would increase or reroute power to magically repair a damaged system. No amount of power in the world will make the phasers come back online if they're blown to pieces.
Most of the systems in question are energy-based, like engines, shields, and energy weapons like phasers/disruptors. In a situation where there is more demand for power than there is power, or where a particular system needs to be as strong as possible (especially shields), diverting power from another area of the ship allows the system to which the power is diverted allows the receiving system to function at full or better-than-full capacity at the expense of the other system.
There are analogies in current-era naval and spaceships; when Appolo 13 was crippled, power became a fundamental constraint, and if they had to use one electrical system it was at the expense of something else (in reality they simply shut down anything they didn't absolutely need to survive in order to keep what they did need working at at least minimal levels long enough to get home). On naval ships, it is possible to push the reactor and engines beyond "100%"; the reactor is capable of more than full military power, but it's not safe. Similarly, "flank" speed is as fast as the engine can be pushed, which is faster than "full" which is faster than "cruise".
I don't think it is right to say that, "Life support, transport beams, warp drive, anything mentioned in the show, really" are "electrical systems".
I suggest it is more like an accelerator / gas pedal in a car: more petrol (power) causes more speed. So more power increases warp drive (and hence ship velocity), shield strength, transporter power, etc.
I would think of the EPS Power Grid (Electro-plasma system) like our current mains water supply today. In that, if you have a greater pressure of water in the pipe, water will flow faster out of connected taps. I think that it would work the same way with the way the ships systems are designed to run on plasma instead of electricity.
