These are custom commercial video systems developed by aerospace companies that specialize in telemetry. For instance, the linked video includes the credit:
The video system was developed by Kayser-Threde GmbH for ESA and DLR and integrated on Ariane by Astrium GmbH. Usage for ATV-4 was financed by DLR and ESA and supported by Arianespace and CNES.
These are two aerospace companies that provide a wide range of telemetry, data capture, video, and other solutions for space flight.
Cameras such as these have been included recently for a variety of reasons. On manned spaceflights, they need to look for safety events, such as falling ice, to know if they need to inspect the spacecraft before return, or if an abort is required. On commercial flights these are included to understand failures better. They are also useful for marketing and publicity, but the bandwidth and cost (not just financial, but space, weight, and power on the craft) have been too high for regular inclusion. Now they are smaller, cheaper, lighter, and require less power to transmit an HD image in real time, so the added data gained from them makes them a worthwhile investment.
More generally speaking, these videos aren't much different than radio video signals used on the earth, with the exception that launches always have continuous antenna dishes pointed at them, even as they rotate around the earth. Creating the video, encoding the data, and sending it via a chosen RF frequency isn't hard, but by the time the launch "ends" the vehicle may be halfway around the world, and well out of sight of the original launch site.
Therefore all data communications are done by first coordinating a set of frequencies that will need to be received on earth, and sent from earth. Then NASA works with other space agencies with receiving and transmission arrays arranged around the earth along the launch path to get time on their arrays. As the space ship moves away from the launch site, they coordinate switchover so it appears seamless to the launch control center, but in reality may have switched several times from one site to another as the spacecraft accelerates.
The live video feed from the moon was done similarly, except the moon travels very slowly, so they didn't have to switch from site to site during the transmission (if I recall, it was all received in Australia and then re-transmitted around the world using other communications channels).
With large receiving dishes, and very powerful radio transmitters on earth (because space, weight, and power are cheap on the ground) you don't need much space, weight, or power for the receivers and transmitters on the rocket. They are powered like any other radio system on the rocket, and probably transmit at a relatively low power mode.