Question: Why are racers on such a quest for weight reduction?

Lighter is Better
Everything on a race car is a big compromise when you change one thing, or one system it effects at least one other thing or system and sometimes more than one. Weight however is the one thing that has an effect on all the cars operating parameters. The obvious start to this discussion is that a lighter vehicle can accelerate faster than a heavier vehicle if the vehicles have equal traction because it has a higher power to weight ratio. An example of this phenomenon would be a 200 hp Mustang that weighs about 3500 pd. can easily out accelerate a vehicle that has 400 hp if the 400 hp is installed in a tractor pulling a trailer full of Mustangs with a combined weight of about 80,000 pd.--- you do the math but we have all seen the results at any traffic light.

Manipulation of Energy
 If we talk for a moment about road racing, we know that they don't just relay on acceleration, road race cars have to slow down and negotiate curves and corners as well. these operating parameters are all reliant on the manipulation of energy. It takes energy to accelerate the vehicle, the energy then must be dissipated to slow down or decelerate - - a heavier vehicle has more energy to get rid of through the braking system so therefore the efficiency of the brake system or it's ability to change the potential energy (weight X velocity) into heat and get rid of it is directly influenced by the vehicle weight. In order to change the cars direction of travel we turn the wheels and the front tires work against the road surface and supply a force to push against the vehicles tendency to go straight - - the heavier the vehicle the energy the tires consume to change the cars direction. And if the tires can't develop enough grip to change the vehicles direction the vehicle has to go slower to a point where the tires can change the vehicle's direction. If you force the tire beyond it's capability it will over heat the compound on the surface of the tire goes into a temperature range that it starts to breakdown or change properties, it then cannot develop the same amount of grip - - - this leads again to a slower lap time.. If you are utilizing aerodynamic devices to help the vehicle negotiate corners - - heavier vehicles need more down force - - more down force is created by more wing which requires more energy to push through the air -- more energy means more horse power etc. 

Up until now we have been talking about overall vehicle weight but it gets more complicated because these same principals are applicable to any of the operating systems of the vehicle - - - continuing with the energy idea then consider an engine with lighter components . An engine uses a significant amount of its power to accelerate its own internal components so it stands to reason that if any of the engines moving parts are lighter and require less energy to move then there is more power available for the vehicle to accelerate. If the wheels stay in contact with the road surface the vehicle has more directional stability and ability to maneuver - - - so if this "unsprung weight" is less and can react to the road surface more efficiently or more quickly to the changes in the road surface and therefore stay in more constant contact with the surface the direct result is the vehicle can be more capably maneuvered as a result of the tire staying on the road to have more net grip the cornering capabilities are higher. 

Developing more Grip
Remember everything is a system --- looking at the overall picture --- if the lighter wheels stay on the road more --- developing more grip then the cornering speeds can be higher --- if the car comes off the corners faster due to the higher cornering speeds it has a head start on the acceleration process that it will go through on the straight away additionally if the engine has more available horsepower to the driving wheels due to its lighter internal components then it can really accelerate the car down the straight-away, because the whole vehicle is lighter and has less energy stored that must be dissipated through the brakes system it is easier to slow down or can brake later for the next corner - - but due to the higher cornering capabilities it doesn't have to slow down quite so much so the cornering speeds can be higher and so on and so on. These total energy savings add up to faster lap times on a race course.

 Another weight related topic is the position of the weight or weight reduction.  The position of the center of gravity or the virtual position of the weight has an effect on the whole picture as well. The weight of each of the components especially large dense parts and their relative position to the center of gravity of the vehicle has a lever arm effect that can play a big part in the way that a vehicle behaves. Cars that have the engine in the rear behind the rear axle VS mid-engine cars VS front- engine platforms all handle or feel totally different not necessarily better or worse just different . . for racing purposes the thinking is still that you want the high density weight as close to the middle of the vehicle as possible to minimize the pendulum effect of the weight on the end of a lever arm - - - F1 cars all have this mid- engine configuration for this exact reason Taking in all of this one can understand why the sanctioning bodies specify minimum weights. 

Auto Racing is Show Business
 Auto Racing is show business and the more equal the cars are presumably the better the show for the paying crowd - - - weight is the easiest rule to specify, measure, and police.