These particles (anyons) live in two dimensions, so their world-lines trace out paths in three dimensions (two space dimensions and one time dimension.) These world-line paths get braided or knotted, just like hair does. And even though your hair gets jiggled around by the wind (quantum noise) those braids remain intact (the quantum information is preserved.)

Edit. For the very bravest of five year olds, I wrote a paper on these things here: https://arxiv.org/abs/1610.05384