Why are these two paths (i.e. the red and the black) homotopic :
Because we cannot break continuously the loop, can we ? Indeed, to take off the loop, we have to cut the path somewhere and then glue it again to get the red path. So we cannot get the red path by deforming continuously the black path, do we ?
Same question if one path has a knot :
It's says that both path are homotopic, but we have the same problem, we can not get a line without cutting somewhere. So we cannot get the red path by deforming continuously the black path, do we ?
If yes, I really don't get this things of homotopy.