The experiment described is an osmosis experiment. Osmosis is the process of molecules (particles) in a liquid solution, gas, or gaseous solution (but not a solid, since the molecules in solid matter are not mobile enough) balancing out in terms of concentration across a (partially permeable) membrane.
Here, the Visking tubing acts as the membrane, which would in reality in nature/biology be for example a cell membrane or the surface of a lung. Water is put on one side of the Visking tubing, and a sucrose solution (sucrose being a particular type of sugar) on the other side.
Sucrose solution is hypertonic to plain water, that is, it is more concentrated/has less water than the plain water. So in the osmosis process some of the water molecules from the plain water move over (across the membrane, or Visking tubing) to the sucrose solution on the other side. Meanwhile, some of the sucrose molecules from the sucrose move over (across the membrane) to the plain water solution. So in fact the two liquids exchange some of their molecules in order that they have equal concentration of both sucrose molecules and water molecules. If there are unequal amounts of liquid on each side, the sides might not end up with the same amount of liquid that they each started with, but will have the same concentration as the other side.
For example, if there is less water in the Visking tubing than a (for example) 10% sucrose solution outside the tubing (in the beaker), the level of the water in the Visking tubing will go down and the level in the beaker up, although both will end up with the same amount of (weaker) sucrose solution.
One could also say that the plain water is hypotonic (has more water content/less concentration of sucrose) than the sucrose solution. Liquids with equal concentration are said to be isotonic.
You could imagine that you have some water (on your side of 'the membrane') and that a large friend (a T-rex?) has a lot of water on the other, but not the recommended dietary amount, and quite a few sweets. For you both to have the recommended dietary amount of water, and share the sweets according to your relative size (the T-rex is much bigger so needs more sweets right?), you would give him some water and in exchange (by 'osmosis', which is essentially 'fairness') he would give you a couple of sweets for you to eat. Everyone's a winner!
This is an experiment sometimes done by a class and sometimes as a demonstration by a teacher. It is used to demonstrate practically the process of osmosis to students.
Osmosis refers to the net diffusion of water molecules from an area of less negative water potential to more negative water potential. That latter statement probably requires some explanation. Osmosis refers solely to the movement of water molecules. If two solutions are separated by a membrane that water molecules will pass through some will move in either direction; however, most will move in a particular direction. This is described as "net diffusion". A solution based on water as the solvent is called an aqueous solution and we measure their concentrations in terms of water potential. The question is not really about that so I do not want to get over complicated about it. In summary, pure water (nothing else present) has a water potential of zero. The minute you had a solute the water potential becomes negative. Therefore, more concentrated solutions are more negative in terms of water potential than dilute solutions.
In this experiment you can use tap water. It will have a negative water potential because it isn't pure water. However, it is more dilute than a sugar solution. In the sugar solution is in the beaker, there will be a net movement of water molecules, i.e. osmosis, from the water, through the visiking tubing, into the sugar solution.
Visking tubing is used because it permits water molecules to pass through but not sugar molecules. If the sugar molecules were moving it would greatly complicate things.
How would you know that any water molecules left the visking tubing? This is done by weighing the visking tubing prior to and following the experiment. As it will lose water its weight should be less at the end of the experiment. I always weigh it. Equally, you could place a known volume of water in the visking tubing and measure what volume remained at the end.