How does a lever magnify force? I understand that energy is conserved when a force is applied to the end of a lever and magnified closer to the pivot point. However, I would like to know how it is the force is transferred and magnified between the atoms of the lever.
 A: The force is not magnified, is the torque that is transmitted; and is not an "amplification effect" of the interaction between atoms. The only role the interactions between atoms play is to hold the lever together. And as long as the forces involved using the lever, are smaller than the internally bounding forces of the lever, it will resist the strain and transmit linearly (not amplified) the applied torque. That is, any flexing applied to it with a torque $d\tau$ will be transmitted to the next element $dl$ in the same magnitude. You can get into the complication of "flexible levers" (not really so complicated) but this does not address your actual query.
The explanation, using an unflexing lever (rigid in this sense) is enough.  It applies to cases where the torque is so small that a the flexing angle $\theta_{flex}$ ($\tau_{flex} = -k d\theta_{flex}$) between two infinitely close parts of the lever is infinitely small. 
The multiplication of the force comes from the difference in arms purely, since torque is transmitted equally throughout the lever. Now the lever works also because you change the rotation point by supplying the pivot point.
To better understand the pivot point, think that you hold the lever in your hand, and in the other extreme there is a weight. Now the pivot point is just the center of mass, which will be closer to the weight. That is why you will struggle to keep it up, since you have to apply a force equal  the weight in order to balance it. This means in terms of torque, restoring the center of mass to the geometrical center.
But if you provide a pivot point, you are fixing the weight of the system there, and forcing it to rotate from that point. Thus maintaining the difference in arms, and a small force applied over the longer arm, creates a torque transmitted unchanged that at a longer arm manifests as a bigger force. 
This is more a geometrical effect that follows from constraining integrity in the lever, and infinite (or big enough) support on the pivot point. Then since the angles have to be equal, angular velocities and accelerations have to be also be equal and only the moment of inertia dictates the force required, which is smaller the smaller the radius.
It looks to me that the referred answer does not tackle the point.
