131.25 is to calc ABV (Alcohol by Volume)
105 is to calc calculate ABW (Alcohol by Weight)
((OG - FG) * 105) * 1.25 = ABV
(OG - FG) * 131.25 = ABV
These formulas are flawed in that they get progressively inaccurate with higher ABV, but are within 0.2% up to around 9%
As to where they come from, they are simply the constants that work for most beers based on the absolute weight of pure water and ethanol.
Edit: just watched that vid you linked. IMO pretty much garbage. For example he says to always read the gravity at the top of the meniscus. This may be the case for some hydrometers but not all and is very carless to be abolute like that as it depends on the surface tension of the solution. Each hydrometer should be tested on distilled water and the bottom of meniscus is your reading then variable surface tension is not a factor.