I have built a variety of Cartesian divers over the years. Some divers have openings for water to go in and out, others are sealed. But the underlying principle is the same: The diver sinks when its weight is larger than the upthrust it receives from the surrounding water.
The diver in the above video has an opening at the bottom. At 0:11, you can see clearly water rising through the opening into the diver to fill up more of the air pocket on the top of the diver. This increases the weight of the diver, and it sinks.
The trigger for the water to rise up the diver was my hand squeezing the bottle. Squeezing the bottle caused the water level in the bottle to rise up and press against the bottle cap, thus increasing the water pressure in the bottle. The increased water pressure forces water into the diver, until the pressure of the air pocket in the diver matches the new water pressure.
In the above video, I used my fingers to directly compress the air pocket on top of the bottle. The increased air pressure means that the pressure in the water also increased. Water is forced into the diver, and the same thing happens.
The diver in the above video is a candy in an air-tight packaging. Needless to say, water cannot not go in and out of this diver. But it can be seen clearly at 0:19 that when the bottle is squeezed and unsqueezed, the candy shrinks and expands. As if the diver is breathing in and out! When the bottle is squeezed, the increased pressure forces on the candy squashes it. With a decreased volume, the candy displaces less water. The upthrust decreases, and it sinks.