Assuming that the video you saw is this one (https://www.youtube.com/watch?v=nk9cTa3UthM
), rewatch the near/far usage of the light, but pay attention to the line of the shadow on the model's face.
Note that when the model is near the softbox, that the line is wider and blurrier, i.e., softer light. But when the model is farther away from the softbox, that line becomes smaller and more defined, i.e., harder light. Sunlight is probably one of the hardest lights of all, and gives a very sharp defined line between shadow and light.
So your understanding is actually correct—the nearer to the light source, the larger the light source becomes, and the softer the light.
Falloff is something else. Falloff is transitioning from light to dark and is about light levels (not quality).
What the inverse square function is telling us is that the amount of light falloff is inversely proportional to the square of the distance. A simple Cartesian graph of the y = 1/x2 function looks like this:
So, think of how high the red line is as being how bright the light is, and how far to the right on the graph, as how far away you are from the light source. (This is why the words "inverse square" are still used, despite the mathiness—to precisely describe how much falloff you have vs. the distance from the light.)
Note that near 0 (i.e., when you're really close), the inverse square falloff is really fast and steep. So, yes, you can get a large transition in light levels at relatively small distances. When you get farther out, the curve flattens, so your falloff isn't as large—but the overall level of light is much lower—something you aren't seeing in the video, because the photographer is compensating for that with exposure settings.