This still doesnt make sense to me. Imagine a continuous line whose positions map to the real numbers between 0 and 1. If I "move" the line over 0.1 wrapping the end back to the beginning (i.e. x2 = (x1 + 0.1) % 1), there will be no points that are in the same position as they were in before.

The theorem only applies to continuous functions.

Brouwer's fixed point theorem only applies to compact convex sets.

Infinite lines don't work, as they are not compact.

Similarly a circle would not work as it is not convex (you're close with your example, you just need to glue together the endpoints to turn it into a circle and make the map continuous).

The line example is what I thought and then I looked the theorem up in Wikipedia. Thanks for pointing this out.

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.