The reason you see this is two-fold:
Usual CRT flicker by principle, CRT frames are not displayed in their entirety at any time, but this is not visible to the naked eye, due to the frequency this phenomenon occurs.
A camera is perturbed by any flicker effect, due to the sampling principle it uses. Aliasing takes place, artifacts are created, and are visible.
It may be surprising, but this effect is the result of not fulfilling the first principle of sampling, known as the Nyquist criterion, sampling must done at a frequency at least twice the largest frequency contained in the signal to sample. This can be illustrated like this:
In this image the green signal is sampled at black points. The sampling frequency is obviously lower than twice the signal frequency. When reconstructing a signal from the samples, the result (red) is a different signal. The analogy with CRT is the invisible higher frequency flicker is sampled by the camera at a too low frequency and the result is a low frequency flicker which is visible.
This is an artifact known as stroboscopic effect. It appears when the real world is sampled (temporal sampling) and we look at the succession of samples. Instead of viewing the continuously changing real world we see a sequence of fixed states. This has two effects:
A fixed state may show things we couldn't see before they were frozen, like a picture which is incomplete at a given time.
The sequence of these fixed states may create artifacts, like adding or removing displacements such as a wagon wheel rotating in reverse direction, or a helicopter hovering with stationary blades.
Stroboscopic effect is one of the many form of aliasing. Aliasing refers to an artifact created when rebuilding an object from samples that not in adequate amount to get a faithful copy. In technical term this means the original object was sub-sampled given it's complexity. This relates in fine to the Nyquist criteria: If a signal contains frequencies up to f, then it mustn't be sampled at a frequency lower than 2*f.
In the effect you see there are two reasons: The way a CRT shows images (actually bands on the screen) and the way the camera captures (samples) images. The effect is the combination of the two.
Image creation on the CRT/LCD screen
In a CRT screen, the image is built by energizing phosphor material on the screen surface, using a small electronic beam, line by line from left to right and top to bottom. As phosphor luminosity cannot be maintained a long time, the screen must be continuously scanned by the beam at some frequency, usually 50 or 60 Hz, but today it's often at a higher frequency.
At a given time, the most bright lines of the screen are the one currently refreshed, and a few lines which were refreshed immediately before, which number depends on the persistence of the phosphor used. This creates a brighter band visible when frozen by photography.
Passive LCD screens are similar: A passive LCD cell (pixel) is made transparent by storing electrons in a tiny capacitor which is unable to keep it's charge a long time, so this capacitor must be reloaded periodically. This is also done line by line, and also create areas of different brightness at a given time.
A human eye also shows a persistence of vision. When our eye sees a bright point, the stimuli is maintained after the point has disappeared. This is why we can see a photographic flash which actually lasts less than a millisecond. For most people, there is no flickering after 40 Hz. The refresh rate of the CRT was chosen in order to take advantage of this persistence. The electricity network frequency (50/60 Hz) was used as this provided a stable clocking reference.
Does that mean we need to send 50 (60) images to the CRT each second? Actually this would have been too much for the radio transmission associated with TV broadcast, and only 25 (30) images were sent each second, first all odd lines, then all even lines. The CRT actually displays the two interleaved halves of an image each 1/25 s. This apparent frequency of 50 Hz is enough to prevent flickering for humans.
This is similar to movies which are recorded at a rate of 24 images per second by the camera and are actually projected twice or more this frequency, using a shutter, giving a rate of 48 shorter images per second or more.
Stroboscopic effect from the camera
So a flickering actually happens on any CRT, but we can't see it with naked eyes. However, a video camera actually samples the reality to create, say 25 fixed images per second.
But what do we see on these images: Bands! The location of the bands on the fixed images varies from one image to the next, like in any stroboscopic sampling:
Sampling a moving object at fixed intervals, source
This is the main reason for flickering screen on video recordings: The illuminated portion of the screen varies during the recording, it seems the screen is flickering (while it's actually never totally black).
Now imagine a case where the camera scans exactly the same number of images than the CRT, but is not in-phase, that is the camera doesn't scan the line being refreshed by the CRT, and it happens the camera scans a line where phosphors have already lost their luminosity:
In such a configuration when the camera scans the next line, another line has returned to black, maintaining the same black gap between camera line and first illuminated line on the CRT. In such a configuration the camera would never be able to record a line, creating black images. As the camera and the CRT are not synchronized this happens only from time to time. In between partial images of some random sort are recorded.
This effect can exist unless the refresh frequency of the CRT is high enough compared to the phosphor persistence, or the phosphor persistence is high compared to the exposure time of the camera (if that were achieved, other problems would appear though).
There are ways to remove partially this effect, like temporal anti-aliasing which consists in blending several images together. This way bands at different positions appear on the same image, though it's difficult to equalize luminosity over the synthetic image created.
Other stroboscopic effects due to a camera
As the camera also scans the sensor matrix receiving the image line by line, lines recorded actually belong to different instants. Lines can't match if the objet moved during the assembly of image lines. Because of this a fast moving propeller can appear completely bent:
Video image made of lines belonging to different times, changing the shape of the blades, source
This image is not a high speed photo freezing the propeller, on such photo the propeller wouldn't be distorted. The distortion is caused by the way the camera scans its electronic sensor, line by line, without freezing the sensor content first.
This is an interesting video of similar effects, like these guitar strings: