The explosion itself isn't the source of the EMP. Its immediate aftermath is.

The high-energy gamma rays emitted by the explosion strike electrons in the gas molecules in the air. (And any molecules on the ground, too, but the air will be what's relevant for our purposes.) This briefly turns the air into a plasma, with free electrons moving at high speeds from the huge kick they got from absorbing a highly energetic gamma ray.

In the lower atmosphere, the air is dense enough that these free electrons cannot travel very far. But in the upper atmosphere, their mean free path (the average distance they can travel without colliding and recombining with an atom) is rather long, on the order of a hundred meters or so. That's far enough that the electrons can interact with the Earth's magnetic field.

As the electrons travel, they start to move in loops under the influence of the magnetic field, as any charged particle would. Since the electrons are traveling at relativistic speeds, this produces synchrotron radiation, in much the same way that a boat speeding through the water creates ripples. This radiation is spread out at all wavelengths of light, and radiates outward from the moving electrons until they recombine. Since the electrons are traveling at relativistic speeds with a mean free path of ~100 meters, this recombination happens in on the order of a microsecond.

Since the electrons emit all their radiation within such a brief time, and since this is happening on the shock front of the original emitted gamma rays, the radiation from the electrons closest to the blast travels essentially along with the shock front (since they're only nanoseconds behind the gamma rays that weren't absorbed). As the gamma rays continue to travel, they knock more electrons free, and the synchrotron radiation from those electrons stacks on top of the synchrotron radiation from the previous ones.

All this radiation adds up, forming a shock wave of light at all parts of the spectrum - that's your EMP. (Or rather, it's the first and most damaging of a couple of unrelated EMPs.)

In a ground-level or lower-atmospheric blast, however, there are two differences:

• All the gamma rays are quickly absorbed by nearby air, within a few kilometers. That means essentially all the gamma rays are absorbed near the ground, where the air is dense. The mean free path for electrons in such dense air is much shorter, so they have much less time to emit synchrotron radiation.

• What radiation is emitted only hits targets within line of sight of the immediate shock wave, i.e., within line of sight of a few kilometers above ground zero. The horizon from a few kilometers up is not that far away, so the effects of the original blast tend to be more important than the EMP at that range.

But if you detonate a nuclear weapon at high altitude, ~half of its gamma rays will be absorbed in the upper atmosphere (since the Earth occupies ~half of its lines of sight, just as the ground and sky occupy half on the ground). Those electrons all get the nice long mean free path, and the EMP is emitted at such a high altitude that the horizon is hundreds if not thousands of kilometers away. You generate a potentially massive EMP that can affect an entire continent, which is far beyond the direct blast range of even the largest nuclear weapons.

There are several types of EMP. The one that people worry about, that goes over a long distances, is the HEMP, or High-Altitude EMP. It is caused by a nuclear weapon essentially detonating in the upper atmosphere, and caused by interactions between the radiation from the bomb and the upper atmosphere itself. The easiest way to think about this is that a nuke detonated in outer space has far more of its energy stay as radiation (and not turn into blast and heat by interacting with the atmosphere), and that radiation is used to "charge up" a layer of the atmosphere in a way that results in the EMP. The physics of this is complicated.

Nukes detonated in the atmosphere also produce an EMP effect, but the high-intensity range of it is pretty limited. Limited to the point that if you have something that can be damaged in that range, it probably is close enough to the other effects to be damaged anyway. So it is less "special" in this way, and would just be part of the general damage you'd have of a place that gets nuked. It is not widespread like the HEMP.

All nukes are meant to be set as airburst, it increases their effectiveness significantly

All nukes also create an EMP but in the thicker parts of the atmosphere it doesn't travel as far and the shockwave greatly exceeds it so it's ignored

High altitude nuclear detonations (100km+ up) create a more wide ranging EMP but if you go up too high it weakens

The nuclear blast fires off gamma rays. These Gamma rays hit air particles in the upper atmosphere and rip electrons off them and send the electrons flying. The electrons then spiral down the magnetic field lines of the Earth creating a very strong very fast moving current that can cover a large distance

In the lower atmosphere the air is thicker so while the electrons still get ripped off they'll pretty quickly bump into another atom and get slowed back down