I don't know if the Germans ever issued a report on the gun in question, but jamming was not unlikely. Gabcik's mission was enabled by the British, who supplied him with their Sten gun. Like other submachine guns, the Sten gun used a short pistol cartridge , in this case 9mm Parabellum; a good choice for the mission, as this was the standard German pistol cartridge. And like many submachine guns, it fired from an open bolt: on being released by the trigger sear, a heavy breechblock with a firing pin was propelled forward by a spring, stripping a cartridge from where it was held by the lips of the magazine, chambering, and firing it. The inertia of the heavy breechblock allowed the gas pressure in the barrel to drop to a safe level, before the breechblock was shoved back into the spring, flipping the spent cartridge case out the side, and the block was sent forward by the spring again. It was very simple and cheap, but the design of the Sten magazine had flaws. It required two staggered rows of bullets to merge at the top into one row, which could allow dirt and fouling to prevent feeding; and, of course, combat tended to make guns dirty. It was easy to wear out the magazine catch if the gun was held by the magazine ( movie-style) instead of at the forend, like a rifle, which would allow it to skew in relation to the travel of the bolt and impeed feeding. And because it fired from an open bolt, it was possible the gun could jam on even taking the first shot. So,the functioning of the magazine was critical, and likely was the problem for Gabcik...as it was for many who used the Sten. And, it should be admitted, was a problem for many semiautomatic and automatic weapons. Another 9mm weapon, the Luger, even when clean and in excellent shape is/was notorious for being very particular as to the ammunition it uses.

He likely did nothing wrong. He had been trained on the weapon before, in Britain, so it wasn't a matter of incompetence or not knowing how it worked. The Sten failed to feed. This type of malfunction is seen across all magazine-fed small arms to one degree or another. But it has specific consequences in submachine guns because it can occur on the first shot.

Like most submachine guns, the Sten supplied to the operation Anthropoid personnel by the British fired from an open bolt. This means that a FtF can occur during initial firing. The bolt has a little nub (which you can feel, if you stick your finger into the chamber through the ejection port) on the front end that serves as the firing pin. When the trigger is pulled, the bolt slides forward, strips a cartridge off of the mag, chambers it, and when the nub hits the primer the primer goes off. Whereas, closed bolt weapons (the vast majority of autoloading weapons, and also bolt-action infantry rifles) perform the loading process before the trigger is pulled. In technical terminology, the system goes "into battery" as part of the firing process, not before.

Incidentally, when you shoot one of these weapons there is a perceptible delay, after you pull the trigger, before the shot breaks. They're not built for accuracy.

If you've ever played paintball, the pneumatic paintball markers offered for rent at many paintball places are in basic function no different from the Sten. They use a heavy bolt that moves forward to slap a gas valve which releases air pressure that moves the bolt back and shoots the paint. In a submachine gun, the gas pressure comes from the powder deflagration. This analogy often helps people appreciate just how dirt-simple these weapons are. Britain started making them during a time of austerity when the Germans were threatening to invade the British mainland. They didn't have the option of spending years to develop a system that was failure-resistant or that required complicated machine tooling. Theoretically, you could have made something like a Sten during the U.S. civil war: that's how simple it is. There is alternate history fiction about that.

u/Bodark43 did a good job of explaining the details of the other issues that the Sten had, but the tl;dr version is that Stens tended to jam, the one most likely used here was the Mk II which was also the most cheaply made, and because of how the Sten works, you can get more and different kinds of failures than on something like a bolt-action rifle. But, they were fairly easy to disassemble and hide, which is why they were selected for use in this operation.

Even still, the backup grenade worked. I always thought that the Czechs had serious guts for pulling this off, knowing what the consequences would be. I can't imagine that it was an easy choice for the decision makers knowing that the consequences would have largely fallen on the Czech civilian population. Which they did.