Where does the extra force generated by a lever come from? Given an object being pulled down to earth by a force of 1000 lb:
If a machine were to lift it off the ground for me, and I grabbed on to the rope before it was released, there is no way in the world I could hold it in place.
But, if I had a sufficiently long (and weightless) lever, I could achieve this with just a finger, with a force of less than a pound.
So where do the other 999 lbs of force come from? What triggers it to be applied now when I apply my finger as opposed to before I did?
(I understand the rules of torque and its equations; I am looking for and understanding of how & why. It does not help to be told that "by the rule, this must occur, and therefore the rule makes sense" - that is obviously circular. And saying, "well, observation bears it out" defends it but fails to explains it.)
To clarify, I'm just asking assuming there is a better answer than "it is a basic law of nature: torque causes force to appear out of nowhere, and cannot be explained in terms of other more basic laws or attributed to some supplier of force". When I ask to understand, I mean as opposed to an axiom, i.e. a starting point of reasoning, accepted but not understood or derivable from other accepted/proven rules. Of course, it's definitely possible that it is an "axiom".
After seeing several answers, I think the following is what is really irking me: Shouldn't there be, intuitively, conservation of force? How can there ever be something from nothing? Unless we define a new type of spontaneous force 'lever force', outside of the known four, then it must arise from the known four. But the known four transfer exactly what was applied to them (along the same line), no more no less. (Applying 1lb of force to the top atom in a stack of atoms will never increase its force on the next atom by 2lb after equilibrium is reached - it will be increased by exactly the 1lb being applied to it.)
In response to the suggestion that the fulcrum provides the force, please consider the following example: A Ferris wheel with a 1000lb load in one seat. It falls straight to the bottom. And ultimately, like is being suggested, it won't fall to the ground, because the center spoke supports it. However, it will hang straight down - i.e. the center spoke is not keeping the object an extra $r$ (where $r$ is the radius of the wheel) feet in the air! Clearly, any time you raise an object above where it was supported by another object, it requires a force equal to that of gravity. A table-top will never assist you in raising the object above the table top, though it will keep it off the ground (should you release it). Think of the table-top as the seat-containing-the-object hanging straight down - any higher is like lifting the object off the table, and should require a force equal to gravity. Except it doesn't. Hence my question.
 A: Jeff, you write: 

And that is what an axiom is, accepted but not understood or derivable from other accepted/proven rules.

In this case, that is simply not true. You can derive the fact that increasing the length of the lever increases your applied force just using the high-school physics expression $w=Fd$. Here's how:


*

*First, note that your question is logically equivalent to the question "why is it easier to unscrew a bolt using a longer wrench?". If you do not understand why they are equivalent, I will clarify with a comment.

*Suppose it takes $E$ Joules to unscrew a particular bolt, which comes loose with one full turn. Since $w=Fd$, if you use a wrench of length $R$, you have $E=2\pi RF_h$. Hence the force you apply with your hand is $F_h=E/(2\pi R)$. 

*However, the bolt still resists movement with the force $F_b=E/(2\pi r)$ irregardless of what shape of wrench you use, where $r$ is the outer radius of the bolt. The ratio $F_b/F_h=R/r$. This is exactly the lever force equation.
Thus, if you accept that $w=Fd$ (as confirmed by experiments), you have no choice but to subsequently conclude that levers allow you to generate extra force. And indeed, in reality one finds they do. It's all self-consistent.
A: First, it's important to differentiate between lifting the weight and keeping it from rotating.  Your finger doesn't hold it up, the fulcrum does that.  All your finger does, is to apply a force to keep it from rotating downward.  
So, ok, why do you need less force the further out on the lever you go?  The answer lies in geometry, the definition of "work", and the law of conservation of energy.  Moving the lever further out by some distance translates into moving it a much smaller amount close to the fulcrum.  So, if you do some work (W = F*d) far out from the fulcrum, you'd expect the same amount of work to be done by the lever (a rigid body) near the fulcrum.  But, to do the same amount of work when d is smaller requires a larger F.  Hence, the force applied far from the fulcrum becomes magnified up close, simply due to geometry and conservation of energy.
I hope that helps.
A: To gain insight into this problem without appealing to "force amplification" etc, replace your finger with a 1 lb weight and consider the center of mass (COM) of the system assuming the 1000 lb weight and 1 lb weight are connected with a rigid and massless beam. 
Assuming the system is free and parallel to the ground with no fulcrum, gravity acts on the COM as if the system were a 1001 lb point weight, accelerating the system without imparting a rotation to the beam.
Now, if the fulcrum is placed under the beam at the location of the COM, the fulcrum provides a counter force through the COM such that there is no acceleration of the COM and no rotation of the beam.
Of course, if the fulcrum is placed elsewhere along the beam, the force from the fulcrum is not through the COM and thus, the COM accelerates and the beam rotates.
So, from this perspective, the question of "where does the extra force come from" doesn't apply.  Either the force of the fulcrum goes through the COM or it doesn't.
Now assuming the fulcrum is placed under the COM, replace the 1 lb weight with the 1 lb force from your finger...
A: The extra force in a lever comes from the distribution of energy over a smaller distance of action.  We can look at it very generally from the perspective of conservation of energy, or the lossless transmission of work through a machine.
When we move something against a resistance, with a fixed force, we are putting in energy and doing work. This work is the product of force and distance: $w = F \times d$.
Devices which use pulleys, levers or hydraulics to generate a larger force all work by "trading" to a smaller $d$ and larger $F$. Since the machine is efficient (little energy is lost in the machine), the amount of work $w$ done on the machine by you is almost fully transmitted to the load. Since the distance $d$ by which the load is moved is smaller, the force is greater.
For instance a lifting machine whereby we push against a force, and move a meter, such that the load lifts by 1 cm, gives us a 100 fold advantage. The tradeoff is that although we moved something by a meter, the load only moved one centimeter. We paid for more force by sacrificing range of action.
We do not have to appeal to energy to see that large forces are generated; another way is to draw free-body diagrams of the parts of the machine, and the force applied to it, and to the load. Balancing the forces requires that a large force must be present on the load.
A: Re: your second edit
No, there is no conservation of force. Forces can and do arise spontaneously. Imagine two balls colliding head on in free space (so no gravity, friction etc.). Suppose they collide elastically and bounce off in some other direction without losing any energy (for simplicity). Also assume for simplicity that we are sitting at the center of mass of the two bodies, so there is zero net momentum. They come in, collide, and bounce off symmetrically. This is just a convenience, you are free to look at this in any other frame of reference. You can approach this kind of ideal experiment with something like an air hockey table, or on a space walk. The idealisations aren't really important here, it's just to keep things simple.
During the collision there are huge forces acting on the bodies to change their velocities. Where did these forces come from? The answer is they didn't "come from" anywhere. Force isn't a substance which is stored inside things waiting to be used up (this is an Aristotlean misconception that many people have without realizing it, and it may be what's at the root of your problem). It didn't come from the energy of the bodies since the collision was elastic, i.e. the energy didn't change. And it didn't come from the momentum of the bodies as the total momentum is also conserved since there are no external forces acting on the system. In fact, after the collision the state of the balls is pretty much exactly the same as it was before, only their direction of motion is different. But the laws of physics don't have any special direction built in to them - they are "rotationally invariant," so there is actually no property of the objects which could be "used up" in this example to create a force.
In fact there is a way of understanding this process without invoking forces at all by thinking in terms of the electrostatic potential energy of the atoms in both of the balls. This is a more "modern," fundamental way to do physics and forces are derived as a consequence.
Force is not conserved, but momentum and energy are. Force is the rate of change of momentum. So you can think of momentum as a fluid. A force is a flow of momentum into or out of a body. (And incidently Newton's third(?) law, the action-reaction law, just says that as much momentum leaves one thing as goes into another. Dead simple.) Asking for a conservation of force is like asking for a conservation of flow. It's just the wrong level of description. Fluid is conserved, not the flow of the fluid. Flows are temporary things which arise as needed to make the fluid go from one place to another, without changing to total amount. Similarly forces arise as a way of redistributing momentum so that the total amount of momentum never changes.
A: That extra force comes from the fulcrum of the lever (the thing which supports the lever on the ground).
This fulcrum balances the downward forces on the lever in total (1000 lb of the object + 1 lb from your finger).
So Archimedes' claim about being able to lift the world with a long enough lever would only be possible if he had a strong enough fulcrum and a support (like a rigid wall) to keep the fulcrum in place.
A: Perhaps it will help you if you reduce this to an edge case and build from there.
Say we have an (infinitesimally thin) 1kg load, resting right on top of an (infinitesimally thin) fulcrum. Do we even need a lever? No, it's balancing on its own. The fulcrum is providing 10N of force, straight up. 
Now move the load an atom's breadth to the right. It's going to fall on its own because the fulcrum isn't directly under it any more, so we'll add a lever to extend the fulcrum's reach. But now there's the load on one side and nothing on the other, so we need to add something on the side opposite the load to keep it balanced. If I make the lever a meter long, do I need to put 10N of force on the other end to keep the load up? Obviously not, I barely need anything (0.1µg?). The fulcrum is still providing the full 10N to keep the load aloft, it just needs a little bit of an extension so it can provide that force directly under it (+ for the speck of dust). 
Now slide the load over a mm. We need to extend the fulcrum's lifting force a bit more, so we slide the lever over too. The fulcrum is still providing all 10N of force we need to lift the load, the lever is still just extending the fulcrum's reach, we just need a slightly larger counterweight (~1g) to balance the torque.
OK, slide the load to 20cm. We need to extend the fulcrum's reach even further, so we slide the lever over as well. There's still an unbalanced torque on the lever, so to keep the load from falling we need to add some more counterweight (2.5N/250g). Despite the increase in counterweight, the fulcrum is still providing the upward force for the load, just from a longer distance. 
We can repeat this process to any lever ratio, and it will always boil down to the same thing: the fulcrum is providing the upward force to keep the load in the air, and the counter-force on the lever arm is only there to keep the torques balanced so the fulcrum can do its job.
Don't think of a lever as amplifying the force you apply, think of it as a way to let the fulcrum do the lifting.
A: There is no extra force.  Draw a free body diagram of the system before and after you push on the lever.  Before, the weight is reacted where it rests on the ground.  After, the weight is reacted at the fulcrum.  The only additional forces are the 1 lb you apply with your finger and the 1 lb reaction to that at at the fulcrum.
A: What does the principle of the lever state?
it states that there is a compatibility condition $\frac{F_2}{F_1}=\frac{r_1}{r_2}$, related to torques (which, transposing, $F_2 \cdot r_2 = F_1 \cdot r_1$ is another form of the conservation of energy).
See a related question for the relation between torque and force.
So the lever compatibility condition (for torques) has the result of direct amplification of the linear force (which is related to Newton's 3rd Law).
An analogous lever principle holds also for transformers (in electronics, electric circuits), by amplifying voltage (but de-amplifying current, since the output power is the same as input power).

A transistor as "electronic lever"

Still another example of a (similar to) lever prinsiple in hydraulics/hydrodynamics is the flow between pipes of different cross-area (the pressure increases).

and the hydraulic press (hydraulic lever)

Indeed there could be a conservation of force principle (under different conditions, for example for the collision of two identical particles on a straight line).
Where does the extra force come from? It comes from the conservation of energy. 
Because the application of force in one end of the lever, assuming different radiuses, has as a result different potential energy (energy based on position). And since for the simple examples of levers, assumed, the forces are conservative (derived from potentials), the rest follow.

A "Judo lever"!

PS  One could say that, as transformers amplify voltage (with the help of the magnetic field), so (mechanical) levers amplify force with the help of another field (e.g gravitational).
A: Quick answer, the ground. Try to use a lever on unstable ground (like mud) see what happens.
A: Imagine you have a balance scale. This is a type of lever. You've got a 10lb object in one pan, and you are pushing down on the other one with 10 lb of force to keep it level. The weight and your hand together are pulling down on the pin at the center of the scale with 20 lb of force, same as if you had two 10 lb weights on each side.
Now imagine it is an unequal arm scale. The weight is 100 lb, and is only 1 inch away from the  pin, and your hand is 10 inches away, still applying 10 lb of force. Since you are pushing down with 10 lb of force, and the object is pushing down with 100, the pin has to withstand 110 lb of force total.
Now make it an even more unequal arm scale. The weight is 5 feet away from the center pin, and you're almost a mile away. The weight is now 1000 lb, you are pushing down with a force of 1 lb, and the center pin is supporting 1001 lb.
If you've got a different kind of lever where you're pushing up, the fulcrum may be supporting 999 lb instead of 1001 lb, but it's still holding up the bulk of the weight, and will break away if it can't handle that much force.
(I don't know why other people are saying there's no conservation of force - it's not called that, but that's what Newton's Third Law is - nothing's accelerating, so all vectors sum to zero.)
A: Of course there is no extra force/energy generated here:
a lever simply does to mechanical work what a magnifying lens does to light and EM energy.
It concentrates energy in a smaller space, so that the local effect is magnified but the total energy remains the same.
