How does a car's low gear increase power while going slower? People have always told me that when a car goes into low gear, it goes slower but puts "more power" or "more torque" on the wheels, which allows it to descend slippery slopes or get out of mud traps more easily.
I've never had a satisfactory answer as to what that actually means. Doesn't more power on the wheels mean they'll go faster?
 A: Power is force multiplied by velocity. The engine power is actually (relatively) constant regardless of the gear. So when people say there is "more power" in a lower gear, it's the common misconception that "more powerful" is "more forceful" but that's only part of the equation.
So if P is constant, then that means if you can combine a large force and a low velocity or a low force and large velocity for the same power. 
When you put it in a low gear, you produce a large torque -- or a large force -- and a low velocity. 
For example, if you are towing a trailer or trying to climb a very steep grade, you need the force to be large which is why you put it in a low gear. If you are on something slippery like snow or ice, a high gear will keep the force at the wheels low so the tires don't exceed the coefficient of friction and spin.
A: It's misconception or misapplication of the term "power". 
Power is force x speed in linear terms, or torque x rotation rate for a rotating system. 
Automotive engines typically produce power over a limited range of rotation rates (customarily measured in RPMs). A given engine will have a characteristic "power curve" and a "torque curve" - basically, these curves indicate  the maximum power and torque available for a given RPM, and can similarly indicate the ideal RPM the engine should be operating at for a given load condition. Away from that ideal, the engine is either unable to accept the load (stalls), will be operating inefficiently, or is under some form of stress which will either cause unnecessary wear or possibly even damage.
The purpose of the transmission is to attempt to match the driving conditions (speed and load) to the characteristics of the engine (torque and power available at a given engine RPM) by selecting an appropriate gear ratio. Selecting one gear (gear ratio) over another exchanges rotational speed for torque (or vice versa).
If you are driving at constant, modest speed on flat road, the ideal gear is probably a "high" gear - one which allows the engine to turn slower because there is little demand on it. If you encounter a hill and want to maintain speed as you climb it, you need to deliver more torque to the wheels, which means you need more power from the engine. Upon selecting a lower gear, the wheels are still turning at the same speed, but the engine is turning faster, which means it can produce the additional power needed by the wheels to meet the new driving conditons.
Now consider accelerating at maximum effort from a slow speed. You apply full throttle in a low gear, allowing the engine to operate at some "preferred" RPM. As the vehicle accelerates, you shift to a higher gear, the vehicle continues to accelerate, you shift up again, and so on. You are always keeping the engine at or somewhere in the range of its maximum power RPM, so on average, the same amount of power and the same torque is being delivered by the engine to the transmission as you run through the gears. At first, that power is delivered to the wheels as high torque, low RPM (you get maximum acceleration). As the vehicle accelerates, and the wheels turn faster, you are shifting up, so less torque is being delivered (and so acceleration is not as great), even though power is still the same, until you reach your desired speed or run out of gears.
Within the capability limits of the engine, shifting into a lower gear allows the engine to turn faster, allowing it to produce more power which the transmission converts into greater output torque for a given wheel speed.
A: I know its an old question but the explanations seem too complex for a layman. I think examples will be apter.
consider holding a heavy object in hand and stretching your hand straight and compare it to holding it close to you. It should be easier for you to hold it near you. That is torque doing its thing. I can say that T = F x r, which is force multiplied by distance but that would confuse the majority of the people.
Now that you understand torque we will apply that to gears, at lower gear the axle connected to the engine will have smaller gear disc size, and therefore it will have more force just like holding something heavy in hand near to you compared to further away. But by decreasing the gear disc size, the wheel rpm will decrease and hence the speed will decrease. When in a higher gear, the gear disc size increases and torque decreases and rpm increases.
Actually, I have skipped explaining how different gear sizes change the speed but that will be a question of its own. I will attach a gif of the gearbox to get the gist of it but I would encourage to search how gears work or gearbox works to understand in much detail.

Image ref:Wiki - Manual Transmission 
An additional link to a video of explanation of How Manual transmission work
A: The equation that power is the product of force and velocity is only valid when the force applied is not used for acceleration but used to counter the dissipative forces the car might face, on a hill the required force is high because the car is doing work against gravity,this increased force is used to do work against gravity but not to accelerate the car,this is why the larger force wont cause the car to move faster.
This is different on a level road,since the only dissipative force is air resistance,the force applied partly accelerates the car,as a result the car's velocity improves yet the engines accelaration drops since the engine power is fairly constant,consequently gears are shifted to the high end
