This was a very interesting video, but unfortunately I cannot think of how this happens. I have also tried to repeat their experiment but was unsuccessful. Here is my analysis of what was shown in the video.

Step 1) Baking soda + vinegar --> Sodium acetate + water + carbon dioxide
NaHCO₃ + CH₃COOH --> CH₃COONa + H₂O + CO₂
(This is the classic volcano demonstration)

Step 2) Cool the solution in the freezer
The first reaction is exothermic (releasing heat) so the solution is cooled back to room temp.

Step 3) Add calcium acetate to calcium bicarbonate
I thought this was a mistake as the first solution made was sodium acetate. They also showed the solution being added to a solid they call calcium bicarbonate. Calcium bicarbonate is always in an aqueous solution so they likely meant calcium carbonate.Either way there is not much of a chemical reaction that would occur here.

Step 4) Add ionized salt
I believe they meant Iodized salt. This salt is sea salt which has NaCl (your common table salt) as well as some potassium iodide (KI).Still - not so much of a chemical reaction - just some equilibrium occurrences with a mixture of salts

Step 5) They boil it up, mixing constantly
I think due to such a large amount of salt, they heated the solution with mixing for everything to dissolve

Step 6) Let it cool
you may have observed that when the solution was poured out it was white and cloudy but in the last frames it was clear.

They use the argument that contact with the air makes substance lose its "polar ability" - I'm lost here, this just doesn't make sense to me. They have added many polar molecules to a solution - there isn't anything in the air that will take that away.

Thinking back to step 3 - the "typo" of using calcium acetate -- mixing calcium acetate with alcohols can result in a moldable "jelly"perhaps they did more than what they showed.

If you're looking for something else interesting involving water I suggest checking out its effects of surface tension and hydrogen bonding. The water molecule H₂O forms a network around other water molecules. This is due to the strong hydrogen bonding between the lone pair elections on the oxygen with hydrogen atoms of other water molecules. One way to demonstrate this is filling a glass with water and then drop by drop adding more water - you will find that it will fill over the edge of the glass (like a muffin top) before eventually spilling over. Another interesting demonstration is to fill an ice tray with distilled water or de-ionized water (the diminished salt concentrations) allows for more hydrogen bonding and when it becomes frozen you will see a small spike of ice that grows out from the center.

The video, as cool as it would be, is unfortunately a Hoax.The video maker used creative photo skills to disguise his use of absorbent polymer water beads (the same material used in diapers to absorb water). They aren't just water, but water in a very swell able plastic mesh.

Just a heads-up!

The first thing that they are doing is making an extremely concentrated salt solution, but they are augmenting that by using the acetate (an organic molecule) while frozen. I don't know for sure, but my suspicion that the purpose is to turn the whole thing into a liquid-crystal phase with the acetate and the salt. It evidently has extremely high surface tension, which is why it forms into those little balls. I can't say exactly what the chemistry of the acetate in there is doing, though, but this is just my guess.

The water balls are cute but a total fluke... The foaming will happen, though.

The video is a hoax.

I could be wrong, but the video appears to me to be a hoax. Here are several reasons why. If you watch carefully, you can see there are objects in the water before he puts his hand in (wavy appearance), the surface of the liquid looks bumpy, the water droplets on his fingertips don't form balls, you can see the deep water move when he's picking things off the top, and when he scoops them toward the camera, you can see that the spheres are already formed before they're even in the air. In other words, this is just a bunch of spheres already in the water.

There is a superabsorbent gel that looks like water. You can buy it at science shops, magic shops, and sometimes supermakets, or online. Superabsorbent beads will expand in water to the size shown in the video. They have nearly the same refractive index as water, so they seem to disappear (just as glass balls do). It's the same material used in absorbent diapers. I suspect the final video just shows beads of superadsorbent gel in water, and the mixing chemicals is just to fool you.