In a regular release, we unweight the outside leg and tip the ski to flatten it while transferring weight to the inside ski (in an early release) or just float in a typical high-performance release. In the weighted release, we relax and flex the outside leg but keep some weight on that ski as it flattens. That's what gives it the name.
In several of the transitions above, you will see that she will lift the old inside foot, keeping the weight on the old outside foot throughout the transition.
This is accompanied by a massive flexing of the old outside ski and tipping it inside the new turn, while still weighted all the time. Flexing is a must for this type of release, to absorb the energy from the turn because otherwise, the skier will be vaulted over the skis and lose weight on that ski.
As the leg is strongly flexed (i.e. you give into gravity), it can also very quickly move the body over the skis into the new turn, just as the skis are tipped into the new turn resulting in the establishment of early angles, without disturbing the balance.
Something similar is known as the "white pass turn", which teaches releasing and transferring without any involvement of the old inside leg. The white pass as a drill, is a more extreme version of the weighted release, where the old inside ski is lifted and the new turn starts well on the new inside ski.
The weighted release is not just a drill, it is a release mechanism in its own right, especially useful in bad snow and often encountered in Giant Slalom.
Also, commonly the white pass turn is done with an extension and exaggerated lifting of the inside ski, while the weighted release is executed by flexing and un-tipping the outside leg/ski, as are all high-performance releases. The slight unweighting/relaxing and flexing of the outside leg will allow tipping it off the edge, while still maintaining some weight on that edge and keeping it engaged.
This type of release could be called a "late release" in contrast to the early release, as it keeps the old ski engaged and pressure on the old ski until very late in the turn, keeping it turning all this time.