Another practical example is that you're guaranteed to have an annoying spot on your head where the hair sticks straight out and you can't brush it down.
That's actually the one that helped me visualize the theorem. If you look at your scalp from above, you can divide all the hairs into "points left" or "points right" and draw a boundary between them of hairs that point neither left or right. Then you can do the same thing with "points up" and "points down." Where the two kinds of boundaries cross, you have a hair that doesn't point up, down, left, or right - it points straight out of your scalp.
That's actually the one that helped me visualize the theorem. If you look at your scalp from above, you can divide all the hairs into "points left" or "points right" and draw a boundary between them of hairs that point neither left or right. Then you can do the same thing with "points up" and "points down." Where the two kinds of boundaries cross, you have a hair that doesn't point up, down, left, or right - it points straight out of your scalp.