Each entity in source engine has two hitboxes - one in quake physics world, and another in havok (or vphys) physics world. This second hitbox is called "physics shadow".

Depending on type of the entity one of the hitboxes is dominant. Each frame results of havok physics simulation for barrels, rollermines, vehicles and etc will be copied over quake hitboxes of entities. For NPCs, rockets and elevators state is copied in reverse direction - quake to havok. 

Player is special. Normally quake hitbox will overwrite havok shadow, but only if its movement is not blocked by any havok hitboxes. Player's hitbox will be copied in reverse - from havok to quake - if (a) havok hitbox is valid (not stuck in quake world), (b) player is touching a vphys object.

Quickclip works, because in vehicles havok hitbox has collisions masked out for everything except pickups (health, ammo). When you exit a vehicle, collision mask for havok physics should be changed to normal, but only if vehicle is present in the world. Barnacle forces you to exit a vehicle, but this process is interrupted as vehicle handle value, stored in Player class, is not valid. This allows you to "throw" havok hitbox through walls by colliding with a vphys object (condition b) and teleport to a new location after it passes throug the wall into an empty space behind it (condition a).

You can see how it works by enabling "physicsshadow_render 1". Blue boxes are physics shadow and red boxes are quake boundary box. When you perform quickclip, you can notice how red box stays in place, but blue one keeps traveling through the wall. Use noclip and timescale to see it from the side.