The first is the eccentricity of the lunar orbit - 0.0549 which means that depending on where in the lunar orbit the eclipse occurs, the moon is closer or further away meaning the apparent size ratio between the Sun and Moon varies. In fact - it's even possible to have an eclipse where the Sun's limb is visible around the edge of the Moon at totality - this is called an annular eclipse. [1]
The second factor is that the Moon is lumpy! The depressions caused by craters and valleys means that totality, as defined as when the Moon completely occults the Sun's disc, varies according to where in the path of totality you are. Parallax effects move the Sun with respect to the observed position of the Moon, so the Sun will peek through the valleys/craters earlier or later depending on your latitude. This lumpiness actually creates one of the most spectacular features of a total solar eclipse, the "diamond ring" effect - I had the rare privilege of seeing this in the 1999 eclipse that crossed south-west England and northern France - a mere teenager but the image is burned into my brain. [2]
Other minor factors come into play which add to why the shape of the polygon changes across the track - the Earth is a spheroid, not a flat plane so the ratio of Sun:Moon coverage varies along the track; also the Moon is still gently turning as it orbits in a manner called libration - while tidally locked to face the Earth, the Moon still rotates marginally faster or slower along its orbit. [3]
The effects of the compensation for totality variation are shown in the second half of the video past ~1:20.
[1] https://en.wikipedia.org/wiki/Solar_eclipse#/media/File:Annu...