Quantum Teleportation

Quantum teleportation is a method of effectively taking a quantum state that exists in one laboratory and destroying it in the current laboratory and later recreating exactly the same — still unmeasured — quantum state in another laboratory that could potentially be very far away.
This is achieved by first creating an entangled quantum state in a laboratory and moving one part of the entangled quantum state to a faraway laboratory. Now let's say a scientist desires to teleport the quantum state |ψ> to a faraway lab. The scientists does a specific measurement on the combination of |ψ> and their half of the entangled quantum state and the outcome of their measurement will be two bits of classical information. They can then telephone over the results of their two bits of information to tell scientists at the faraway lab how to do a measurement on their half of the entangled quantum state, which will recreate the quantum state |ψ> at the faraway lab, effectively teleporting it. This is an important result in quantum mechanics, especially in regards to quantum computing.
The name is misleading in that it does not create an efficient means of transportation via teleportation — something like the teleporters from Star Trek i.e. a conventional teleporter — where macroscopic objects like humans (composed of 7x1027 atoms) could be teleported to an arbitrary place.
The comic jokes with the fact that news reporters wish to get a story about Star Trek teleporters and the scientist is angry that there is no interest in his quantum version — the reporters are even disappointed when they write their story. The last panel appears to indicate this scientist not only has a quantum teleporter he also has a "regular" teleporter of the type the reporter is asking about, which would be a gigantic news item, as it is one of the biggest challenges of this century (and possibly future ones as well), to build one. He uses the conventional to escape to The Bahamas.
The whole method of quantum teleportation is predicated on being able to first create entangled quantum states and then transport, by conventional means, one-half of the entangled state. Only after this step, could you then destroy the shared entangled quantum state, to "teleport" a different quantum state to the new location.
Quantum teleportation is deeply related to Bell's theorem where it's shown that quantum mechanics is incompatible with the idea of local hidden variables and which has been experimentally demonstrated (though a few very small loopholes still have not been conclusively ruled out). Explaining "it's a particle statistics thing" is a great explanation of the related Bell's theorem experiments, which demonstrate quantum entanglement which is at the root of quantum teleportation. In these experiments, physicists take an entangled quantum state move it apart and then randomly decide which direction to measure each side of the quantum states. Through a statistical analysis of the results, you can demonstrate and measure each entangled particle in a randomly chosen direction. The statistical correlations between the particles are consistent with quantum mechanics and inconsistent with any local hidden variable theory; however this instantaneous wave-function collapse does not break special relativity as wave-function collapse does not allow communication of any information. Instead you can just analyze the correlations after the fact and compare the hypothesis of local hidden variables to the inconsistent hypothesis predicted by quantum mechanics, and verify the quantum mechanical prediction.
In the comic, Cueball explains to the reporter that quantum teleportation isn't what reporters build it up to be, before Cueball successfully teleports by switching his teleportation machine from quantum teleportation to regular teleportation.
The title text refers to the controversial 1940s Soviet Experiments in the Revival of Organisms video that depicts a dog's head being cut off and revived. The film is controversial in that the footage is often perceived as being staged, though the Soviet scientist depicted in the video was attempting these sorts of experiments and this research eventually led to the first Soviet open heart operation in 1957. Another instance of crazy headlines that may not be as interesting to the public when first explained — but may be an important step on the way to some type of practical use.