Elisabeth, although it is difficult to accept, at the end of the video Brian stress the difference between measuring and seeing. The shrink of the taxicab lengh is not only what you see but also what you measure, so it is more than an optical illusion. In that example, the twisting of the cab would be an optical illusion but the shortened lenght would be real “from your perspective”. But of course, if you think that the only “true lenght” is the lenghh measured at rest, this “true lengh” or “proper lenght” does not change, and you could calculate it from the lengh that you measure using the ecuations of relativity.