Without a reference to that video we're left making assumptions.
So one possibility is that the video glossed over the difference between pixel-density between the two types of sensors, which has two effects. First, the smaller sensor, assuming sufficient resolution, may actually be much higher pixel-density than the larger sensor. What this means, is that for a given distance between photographer and subject and the same focal-length, the fine detail on the smaller sensor might be higher resolution. If the photographer cannot properly frame a faraway subject and ends up only using a small portion of the total captured image via crop, the smaller sensor will result in finer detail of the used portion of the image. This has advantages for wildlife photography since it means a 300mm zoom lens will act more like a 450mm on a Nikon, a 480 on a Canon, or a 600mm on a Micro 4/3.
Another possibility is that as sensors have become finer and finer resolution, optical performance flaws in lenses become more apparent. If a lens manufactured for a full-frame sensor has issues with softness, vignetting, or chromatic abberation in its periphery on a full-frame camera but produces quality results in the center, using a smaller sensor that naturally crops-down to that center might result in continued life for that lens if users of smaller sensors value the results even while full-frame users might not.
That last paragraph said, it's also possible for a smaller sensor to reveal optical performance flaws in a lens at its center, since that smaller sensor has higher pixel-density. This is part why I watch a lot of lens reviews and part why I don't like buying lenses without the opportunity to test them first, especially if they're more than pocket-money.
This article describes the math comparing pixel density between 36MP full-frame and 24MP APS-C. In a nutshell, the lower-resolution APS-C sensor has a finer dot-pitch than the full-frame sensor, even with the total resolution advantage the full-frame camera enjoys. It's not as pronounced as it would be if the two cameras had the same resolution though.
All that said, I would take full-frame over APS-C if I could afford it. APS-C's strengh is its good performance with relatively low cost, size, and weight.