On the original video, it's always hard to say how deeply into an explanation does one need to go. This video contents itself with comparing atmospheres to clothing, and calls that "the greenhouse effect." So OK, the greenhouse effect can indeed be thought of as a slowing of the escape of heat, which is also what clothes do. So one must decide if that analogy is enough, or if a little more detail is required. The video makers decided not to include the importance of the fact that the heat for the Earth comes from the Sun originally, unlike the heat escaping from our body that we generate metabolically. Again, one always has to decide what's important.
But if you really want to know what the greenhouse effect actually is, it is more than just having an atmosphere. It is having an atmosphere that contains "greenhouse gases" (i.e., not nitrogen or oxygen), which are gases that are transparent to the visible light that brings the heat in, but opaque to the infrared light that lets the heat out-- something clothing does not do. One reason this distinction is important is it explains why carbon dioxide is important to the greenhouse effect on Earth and to global warming, even though it is a trace component of the nitrogen+oxygen atmosphere.
Diving even deeper, the temperature of a planet also depends on something called the "scattering albedo", which has to do with how much visible light is reflected away from the planet before it is ever absorbed as heat. A high scattering albedo will reduce the temperature of a planet, even before you consider any greenhouse effects. I suspect differences in scattering albedo is what explains the temperature difference of Neptune and Uranus, though it is probably explained somewhere. (I did a quick search, and the first hit
https://www.hull.ac.uk/work-with-us...hat-is-the-coldest-planet-in-the-solar-system points to the possibility that whatever kicked Uranus over on its side might have also caused the interior to cool, and a larger part of the surface temperature of the outer planets is due to escaping internal heat.) I actually think it would be fair to say the relative temperatures of Uranus and Neptune are not completely understood at present. (We don't even know why their color looks so different.)