Why do levers work? If we have 3m beam with a fulcrum 1m from one end, we will find a balance by having a 1kg weight on the long end of the beam and a 2kg weight on the short end.
Similarly, I will need much less strength when I old a weight next to my chest, than if I stretch out my arm.
I understand this, as I understand the maths behind it (W = Fd...), the conservation of energy... But why is it that a weight farther from the fulcrum exerts a greater force than one closer to the fulcrum?
 A: My uneasiness about the operation of levers was that it can't surely be by some mysterious action-at-a-distance that the forces, acting at separate points, are 'balanced'. It might help you, as it helped me, to think about the forces at play INSIDE the lever itself. The top 'layers' of the lever (assumed horizontal) are under tension and the bottom layers are under compression.
The analysis is particularly easy for a lever in the form of a 'Warren truss': a structure of equal-length rods, pin-jointed to each other to form a horizontal row of contiguous equilateral triangles with alternate ones inverted. [The truss looks like two long horizontal rods (so-called upper and lower 'chords') with a zigzag of rods between them.] Draw unequal weights hanging from two of the lower pins, one either side of the fulcrum at distances that ensure balancing.
If you consider the vertical force components of the rods acting on the pins, you'll discover that the slanting members of the truss are alternately in tension and compression. If you then consider the horizontal components of the forces on the pins from the rods making up the top and bottom 'chords', you'll find that these steadily increase as you go towards the fulcrum from the load. And equating horizontal  forces on the central pin from rods either side at the fulcrum gives you the law of balancing – without using the concept of a moment or torque!
A: The fulcrum is an additional source of 'force'. A lever is constructed in such a way that no matter what happens, the fulcrum stays in place. This means that the total sum of forces must be zero, otherwise the lever would start accelerating. Below I show a lever with a certain force $F$ on the left side. By the law of levers, it exerts $2F$ in the opposite direction. We can now reason that the force exerted by the rod on the fulcrum should also be $F$ in the downwards direction, in order to get a net zero force.
This doesn't explain the full story though. It doesn't explain why a longer arm of the input causes a greater force at the output of the lever. For this we need the concept of torque. The torque around a point (a torque is always defined with respect to a point) is given by $\tau=r\times F$. A given amount of input torque can only move the output if the countertorque is not too great, i.e. if $r\times F$ at the output is not too large. So this means either a short moment arm and large force or vice versa. This explanation might not be as intuitive as you've hoped but hopefully it helps at least a bit.

