I shall try to tackle this interesting question using a graph, though I suppose this proof is not quite formal.
First, I shall only concern the "tail" of the worm that is originally located at $[0, 1]$, because this is the part where a point exists without changing its position (if any).
Next, I create a graph as follows:
The $x$-axis denotes the original position of the worm, $y$-axis denotes the new (twisted) position. Hence, the tail of the worm will be shown as a red dot, while the part of the worm $1$ unit from the tail is marked as the purple dot. Note that the red dot must have a coordinate of $(0, p)$, while the purple dot must have a coordinate of $(1, q)$, where $0 \leq p \leq 1$ and $0 \leq q \leq 1$.
Now, if $p = 0$ or $q = 1$, it means the red dot or the purple dot will lie on the blue line. This blue line is actually $y = x$. In other words, if the red dot or purple dot lies on the blue line, at that point, the original position equals the new position, so there exists a point that the worm has not changed its position.
If $p \neq 0$ and $q \neq 1$, then, just like the figure above, the red dot must be above the blue line (area A), while the purple dot must be below the blue line (area B). The body of the worm is continuous, meaning a continuous curve has to be drawn to connect the red dot and the purple dot. So, it must intersect the blue line, meaning there must be a point that the worm has not changed its position.