By what mechanism does a solar flare overload electronics?

How does an EMP actually generate voltage surges in conductive materials?

There are several distinct kinds of EMP, but they affect bodies via the same mechanism - electric force on electric charges.

The electromagnetic wave near the nuclear explosion is predominantly gamma and by itself probably could not induce large current or voltages, because it is incoherent; field of the wave at different points is of different direction, varying on microscopic scale, which would make the electromotive force in macroscopic bodies erratic and ineffective. But there are mechanisms in Earth's atmosphere by which the gamma radiation turns into more coherent radio frequency radiation, due to air and Earth's magnetic field.

In a simplified picture (assuming the EMP is coming from relatively far away, and ignoring contribution of other radiating bodies) the electric field of the wave at any point oscillates in the plane perpendicular to the line of wave propagation. The direction of electric field can be otherwise random and can change in time.

If the conductive body is long (say, a 1m long rod) and aligned with that direction at some point of time, it will experience strong external electric field across its length, let us assume 1 000 - 10 000 V/m (depends on how close we are, how strong the bomb, time after detonation, etc.). This will move the electrons in the body and make the voltage between the endpoints rise very quickly close to value 1 000 - 10 000 V/m to counteract the external field inside the rod.

Not much else may happen to a thick metal rod placed alone. But consider what will happen if instead of the rod, there is some thin wire, or even multiple mutually isolated wires in the same position. The wires can melt from the high induced current if they are thin enough. If not, the high voltage between the conductor and other nearby bodies may break down isolation sheaths and sparks will occur. Large discharge current may flow where it usually does not and equipment will be damaged.

How does this effect scale with the size/cross section of the conductor?

If "this effect" is just voltage between two endpoints, for a straight conductor without turns, the only important factor is distance between these points projected on the direction of the electric field of the EMP. The farther the points, the higher the voltage that can occur between them.

If the conductor is a wire with lots of turns (a solenoid or a planar inductor), the more turns the higher voltage, also the bigger the turns, the higher the voltage.

The voltage created will have less of a damaging effect on thicker wires or wires / equipment with thicker electric isolation.

Electronics often contains field effect transistors (FET) which are very sensitive to even small voltages on their terminals - voltage going just a little above the prescribed operating values can destroy such transistor. So ordinary electronics is likely to be vulnerable to strong EMPs, despite the fact distance of its terminals is relatively small. For a chip with terminals 1cm apart, if positioned wrong, the induced voltage by the external field given above could be 10 - 100 V and without special protection, that will usually destroy the fine circuits or transistors inside.

For general info.

The explosive heat of a solar flare can't make it all the way to our globe, but electromagnetic radiation and energetic particles certainly can. Solar flares can temporarily alter the upper atmosphere creating disruptions with signal transmission from, say, a GPS satellite to Earth causing it to be off by many yards. Another phenomenon produced by the sun could be even more disruptive. Known as a coronal mass ejection or CME these solar explosions propel bursts of particles and electromagnetic fluctuations into Earth's atmosphere. Those fluctuations could induce electric fluctuations at ground level that could blow out transformers in power grids. A CME's particles can also collide with crucial electronics onboard a satellite and disrupt its systems.

If there are relatively simple (high school calculus or below) numbers and equations to model these effects and phenomena, what is the 'power output' of a solar flare? (Bonus points for HNEMP, but I'm mostly interested in solar flares.)

Just to clarify, any EMP directly from a solar flare would be negligible at Earth. The phenomena that impacts the Earth's magnetosphere causing electromagnetic disturbances in our power grids are called coronal mass ejections (CMEs). The processes responsible for both solar flares and CMEs can generate high energy particles called solar energetic particles (SEPs) which can damage spacecraft or injure astronauts. Solar flares can generate gamma-rays and just about every other radiation on the electromagnetic spectrum but the only ones that are directly disruptive are the radio emissions which can interfere with spacecraft communication on the sunward side of Earth.

CMEs, in contrast, are large blobs of plasma that often move supersonically away from the sun. They generate shock waves and accelerate particles to high energies (also a source of SEPs). When they impact the Earth's magnetosphere, they cause a large $tfrac{ dB }{ dt }$ which can induce large currents in power lines, pipe lines, etc. at high latitudes (i.e., $tfrac{ dB }{ dt }$ increases with latitude due to the geometry of a dipole magnetic field).

The impact of CMEs on the magnetosphere also drive strong magnetic reconnection events on the sunward (dayside) and anti-sunward (nightside) side of a magnetized planet. This results in the energization of large numbers of particles within the magnetosphere, some to relativistic energies. Typically this involves a large injection of high energy electrons from the geomagnetic tail toward Earth, which can damage spacecraft either through penetration of the bus to the electronics or anomalous charging of the bus resulting in arcing and ablation.

The impact of CMEs on the magnetosphere can also induce the loss of radiation belts either from precipitation into the atmosphere or the compression of the dayside magnetosphere causes the boundary to move inside the region where the radiation belts normally exist. This is typically followed by a sudden refilling of the radiation belts with even more energetic particles than were previously there (if things were quiet for a few days prior). These relativistic particles penetrate spacecraft and can directly damage electronic components.