The video explains it rather well, actually.
Let's assume that socks and shoes move the same amount of charge. It is a tiny amount of charge, which gets stored in the capacitor formed by our body.
This stored charge gets discharged across the resistance from the body to the surroundings.
The lower the resistance, the faster the discharge. If the resistance is low enough, the charge wont even create a noticeable voltage change in the capacitor, because it is discharged faster than it is created.
So, if you are wearing shoes, the resistance is big enough to get a noticeable charge stored up.
Wearing socks, the resistance is lower (thinner layer not made of a very good insulator), and the charge dissipates as soon as it is generated.
The creation of static electricity is called the tribologic effect.