cl_lagcompensation is a client-side variable that is active by default and basically tells the server to notice what the conditions were from your perspective when you shoot at someone (not counting projectile weapons) to decide how to assign damage. It allows players to aim for the models of their targets and be confident their shots will register as hits. The downside is that players can sometimes receive damage that doesn't jive with what they see on their screens, since lag compensation depends more on the attacker's perspective. This applies to more than just snipers (e.g. face stabs anyone?).
In many other games, leading your shot is required to hit anything at all. For many players used to this perspective, disabling lag compensation allows them to play "normally."
The purpose of the video was to demonstrate what lag compensation does and how players can receive damage that doesn't jive with either their or the spectators' perspectives. Using the command cl_lagcompensation (0 or 1) does not show the hit boxes. Disabling it forces you to shoot at invisible hit boxes, which tends to be a major disadvantage for all but the most hardcore, old-school fps players who can't stop themselves from aiming that way.