Up to this lecture I understand that there is no universal notion of space and time,everything is relative!!! But what next? What will happen if we know these facts?I mean what is the physical implication of this thing?

Not sure about this, to me it seems that the apparent effect of motion on space is the contraction of space in the direction of motion. To me it's more like an optical illusion, because the actual length of, for example, the train measured by someone on the train is the actual length of the train, everything else is an optical illusion due to motion, so in a sense the special relativity calculates the optical illusion of the contraction of the moving train.

How about taxicab wheels at velocities near c? The driver thinks the angular velocity to be constant. The bystander thinks the angular velocity is not constant due to severe deformation of wheel structure. So the driver thinks the wheel angular velocity is constant and the by stander thinks there is acceleration present.

@Elizabeth Why do you think the length of the train that is stationary relatively to you is the actual one, while the length of the train that is moving relatively to you is an illusion?

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.