Explanation

The lumps of potato make a good airtight seal against the inside of the tube, because as they are jammed into the conical hole, they are compressed. This means that if you push one of the lumps along the tube, the gas between them is compressed, increasing its pressure.

The pressure increases until the force it exerts on the second lump of potato is high enough to overcome the friction that is holding it in the tube. As friction drops when something is moving (this is why it is often much harder to start pushing something than keep it going) there is now a very large pressure force on the potato, acting almost unopposed by friction, so the potato accelerates out of the tube very rapidly.

As you push the lump of potato down the tube, you compress the air in the tube. Eventually the pressure builds up enough to overcome friction, and the second lump is fired out.

This is based on the the same principle as how toy spud guns work, but they use something more engineered to compress the air. Air guns also work in a similar way; pulling the trigger releases a large spring which pushes a piston down a cylinder. This compresses the air behind the bullet until it overcomes the friction holding it there, and then it fires down the barrel.

In fact, based on this same principle, there are designs for a gun used to fire projectiles into space. The prototype was called

SHARP (Super High Altitude Research Project) and worked by using a methane-oxygen explosion to fire a piston down a tube. This then compressed hydrogen to enormous pressures, which forced the bullet/satellite up the barrel. Compared to current rocket technology, only a third of the rocket's weight would be taken up by the equipment used to project it, allowing it to get into low earth orbit far more efficiently than using present rocket technology.