Something almost analogous is if you imagine two circles: one smaller and one larger in 2D. Let's say the smaller circle is inside the larger. It's impossible to move the smaller one outside smoothly without intersecting the larger.
In 3D of course this is trivial, just move the smaller circle up away from the 2D plane, then over the larger one and down on the plane again.
To tie it in a bit tighter with OPs article, I would just add that if we could only perceive two dimensions, the inner circle would disappear as we "move the circle up and away from the 2d plane".
It would only reappear to us once we moved it back down into our plane of perception, safely outside of the other circle.
The relevant bit from OPs article that this is analogous to is:
> "What would we see if you watched this happen in real life? Since we can't see anything outside our 3D slice of 4D space, from our perspective the moving (green) loop would disappear, to reappear later in the unknotted position."