Why is it easier to apply torque via short bursts There are two popular tools I use to apply torque to a fastener (bolt, screw, etc.): an impact driver and a drill.
The drill is a motor hooked up to some gears and eventually a bit that fits over the fastener. If I want to apply 40 lb/ft of torque, I feel as though I have to brace myself for that amount of torque, like by using both hands and my body.
The impact driver is a similar motor to the drill, but there is a spring-loaded mechanism that applies the same amount of energy, but in short bursts rather than continuously. I can easily apply 40 lb/ft of torque with my wrist barely moving; using two hands or bracing myself doesn't really make a difference.
Why is this the case? Why is there no equivalent force on my wrist when using the impact driver?
This may be similar to using a hammer to drive a fastener into the ground: if I generate force by swinging very fast with a hammer, why isn't there an equivalent force that lifts me off the ground?
 A: the reason you can sustain torque bursts easier with the rotary impact tool than steady torque loading from the nonimpact tool with the same average power rating is the rotary inertia of the impact tool reflects most of the impact shock into the tool bit before it can get transmitted out of the tool and into your hands (i.e., it is inertially clamped). Demolition tools like pneumatic and electric jackhammers operate on the same principle. 
A: Sorry to revive this old topic, but all the answers were slightly imprecise and I think I can add more. The best way to understand this phenomenon is with angular impulse. Angular impulse is defined as torque integrated over time:
$$ L = \int{\tau dt} $$
In the case of the drill, the torque is applied continually, and so you feel the reactive torque equal to the torque applied to the fastener. For the impact tool, the force is applied over a very short duration, and so the total impulse is very small.
In order for the impact wrench to not start rotating, you still need to apply some torque continually to counter the angular impulse, such that the net angular impulse over time is zero. The impact wrench actually applies the torque during a very small fraction of the time. There must be enough rotational inertia in the tool, such that each individual impulse does not significantly accelerate the tool. In that case, you can counter the angular impulse (so the net angular momentum stays zero) by applying a continuous but much smaller torque in the opposite direction. The tool will vibrate with each impact, but on average will remain stationary.
For example, let's guess that an impact wrench is loosening a fastener by applying 100 ft-lb of torque with impacts such that torque is actually applied 5% of the time. Let's say the tool does ten impacts per second, and each one lasts 5 ms. So each impact has 0.5 ft-lb-sec of angular impulse (100 ft-lb * 0.005 sec), and you get ten of these every second. So, on average, the impulse applied is 5 ft-lb-sec per second. So you can comfortably apply only 5 ft-lb with your hands to hold the tool steady.
In practice, the tool works by converting the momentum of a hammer into torque. So the angular impulse is actually applied at a constant rate, and the counter-torque you need to apply to hold the tool steady also stays constant. The impulse from each impact is applied to the fastener over a shorter and shorter duration as the torque gets higher. This is why you need to be very careful tightening fasteners with these tools -- you get zero physical feedback of how much torque is actually being applied.
A: To get something to turn you need to apply a minimum amount of torque to overcome friction. The impulse of a collision (eg swinging a hammer at the lever) enables you to apply a high torque for a short time, when a constant push with the maximum stationary force you can provide is less than the required minimum torque. The same principle of expending additional energy to increase the maximum applied force is used in the pile driver and jackhammer.
After a few blows the friction force has reduced sufficiently to enable you to continue applying a constant force. This is more efficient than swinging a hammer.
The mass of the hammer is concentrated at the head. The impact then occurs close to the centre of percussion of the hammer. Minimum reactive shock is felt at the pivot (your hand) when impact occurs at the centre of percussion.
