Einstein's equation does not state that you will not be able to see the light when running along side it. Instead they state that the speed of light will have the same finite value no matter the reference frame.
Now, since the speed of a ray of light is finite you would be able to see it as it is passing by you (assuming that there is some kind of scattering of the light which is sending some in your direction). In the case of the video they are taking a series of ultra-fast snapshots and putting them together.
In the case you would ask how the light can keep moving at the same speed when you are moving at near light speed then the explanation lies in time dilation. The fact that your own time slows down as you speed up. One second of your time could be two seconds for a non-moving observer. Hence your individual observations of the speed of light can be compatible.
Notice, that, as you approach the speed of light, your 'seconds' become longer and longer until your time will be practically frozen compared to a non-moving observer. To you the seconds will still seem to last the same amount of time. Due to this what you actually 'see' would be hard to answer.