1642+690 is the type of astronomical object that is identified by its coordinates. The first four digits refer to the right ascension, and the plus sign and the last three digits refer to the declination. This object is a quasar, which is a galaxy with an active galactic nucleus (AGN) that produces huge amounts of radio emission. An AGN consists of a supermassive black hole that is millions of times more massive than the Sun, a disk of gas and dust falling into a black hole, and jets of gas that appear above and below the black hole. These jets are caused by gas that does not fall all the way into the black hole itself but instead get swept away from the black hole towards the poles by very strong magnetic fields. These jets contain lots of ionized gas, which means that they contain a lot of free-floating electrons that tend to oscillate within really strong magnetic fields in the jets, and this causes the jets to produce a form of radio wave emission called synchrotron emission. 1642+690 and many other quasars like it are sources very strong radio wave emission, even though many of them are very far away from Earth. (In the case of 1642+690, we are seeing it when the universe was about 6.7 billion years younger than it is today [1,2,3,4], although that does not quite to a distance of 6.7 billion light years because of various cosmological effects that are too complicated for this podcast episode.)
The radio wave emission from 1642+690 is so strong that advanced radio astronomy techniques can be used to make images of the jet from the AGN on very small scales. In the radio images, the core of the AGN looks like a very bright dot, while the jet of gas appears as a stream of fainter blobs to one side of the core [1,2]. What's really cool is that astronomers have been able to watch the blobs of gas in the jet move over time [1,2]. It isn't moving so fast that astronomer can see it changing while they are making pictures of it, but it is moving fast enough that, if they take a picture then wait a year and take another picture, they can see that the blobs of gas in the jet have moved over the course of the past year. What's really freaky, though, is that the blobs of gas in the jet look like they're moving several times faster than the speed of light [1,2]. That's not possible according to the laws of physics.
What's actually happening is that the AGN is forming dilithium crystals that are powering the blobs of gas to warp speeds... No, wait, that's a script for a badly-written Star Trek fanfic. What's actually happening is that the blobs of gas are moving at speeds that are a fraction of the speed of light but, because of how we view the jet from Earth, it looks like the jets are moving faster than light. I'll explain.
So, first of all, when astronomers make images of the 1642+690, it looks like the jet is moving to the side, but that's because astronomers are making two-dimensional images of a three-dimensional object. The jet is actually pointed at an angle that is almost but not quite aimed directly at Earth. It just looks like it's moving sideways in astronomical images.
The gas in the jet is actually moving quite fast; it's probably moving at close to but just below the speed of light [1,2]. To be clear, radio waves and light waves are both forms of electromagnetic radiation that travel at the speed of light, and nothing can travel faster than that. For everything in our everyday lives, the electromagnetic radiation travels much faster than the objects that emit or reflect that radiation, so we can instantaneously tell where something is by looking at the light or radio waves from that object. In the case of the jet in the quasar 1642+690, however, the gas is almost keeping up with the radio waves that it is emitting because the gas is travelling at nearly the same speed as the radio waves. So, when a blob of gas is ejected from the AGN at the center of 1642+690, it produces radio waves and then travels right behind those radio waves. Moreover, as the blob continues to travel away from the galaxy, it emits more radio waves and then travels right behind those radio waves. What happens on Earth is that the radio waves from when the blob was near the black hole and when the blob was far away from the black hole arrive at nearly the same time, which gives the illusion that the blob is moving sideways at much faster than the speed of light.
This phenomenon is called superluminal motion, and when this was seen in 1642+690 in the 1980s, it was only the ninth quasar where anyone had seen this [1]. Today, its more common for astronomers to see this effect, but thanks to quasars like 1642+690, they now understand what's going on.
So, the three major lessons to remember from this podcast are that 1642+690 is a quasar that is ejecting a jet of gas that can be seen in radio wave emission, that the jet looks like it's travelling at faster than the speed of light even though it isn't, and that it's very likely that multiple people have written some Kirk/Spock fiction involving active galactic nuclei, including possibly 1642+690.