Otto has it essentially correct. The concept was that the gun fired a shot similar to a modern sabot round. That is, a round that had a soft ring of material surrounding a hard sub-caliber shot. The difference was that in the taper (or squeeze) bore gun the soft outer ring was squeezed down by the barrel taper until it reached the size of the hard core shot. This allowed for more velocity through a larger volume of gas to push against the shot.

This system was tried on four guns during WW 2; three German, one British.

The first was the 2.8cm S PzB 41. It squeezed the shot from 28mm in diameter to 20mm at the barrel with a 1402 m/s muzzle velocity. This gave it about half again the penetration of a conventional 20mm cannon.

The second was the 4.2cm le. Pak 41. This looked like a 3.7cm Pak gun except with a longer barrel. It used the same principle with the shot going from 42mm to 30mm at the muzzle. The muzzle velocity was 1265 m/s and it had about double the penetration of a 37mm gun and almost equaled the performance of the 50mm Pak at out to about 500 yards.

The third German gun was the 7.5cm Pak 41. This was Krupp design (for obvious reasons given below), and only 150 were produced. This gun fired a 75mm round that reduced to 55mm at the barrel. Its muzzle velocity was 1124 m/s and it had a penetration almost equal to the 88/71 on a Tiger II out to about 1000 meters. A major problem was that the gun had a barrel life of just 400 rounds at most (about a tenth of a normal cannon of that size).

The problem for both of these was first that Krupp through its subsidiary Hartzmetallzentrall controlled all of the tungsten carbide production in Germany. This created an artificial shortage and raised the cost of ammunition substancially for these guns through the monopoly this company had. This made them uneconomical to use as well as making ammunition scarce.
The second problem was that the barrels wore out very quickly in use. The combination of pressure and velocity were very hard on them. The result was that only a relative handful were built and those that were saw relatively little service before being scrapped.

The last squeeze bore weapon was the Littlejohn adapter for the British 2pdr gun. This came out late in the war as a means to give a bit more punch to armored cars still using the 2pdr. It was a bolt-on assembly that lengthened the barrel of the gun and worked on the squeeze bore principle. The reduction was far less radical than in the German guns only reducing the shot to about 35mm. But, it precluded using regular shot or HE rounds so it saw little service. A few were also fitted to US 37mm guns on armored cars in British use as these could still fire their regular rounds using it. It helped a bit with muzzle velocity but not significantly.