Please note that this is mostly speculation since I don't have access to the full article.
Case summary in common English
A hairball (medical term: Trichobezoar; link to image with a warning for disgusting content) was blocking the small intestine of a cat. The vets inserted an endoscope to diagnose the cat (from the abstract of the paper it's not clear whether they inserted the endoscope through a natural orifice or an incision) and they could see the hairball, but couldn't reach it with their instruments to remove it. So they flushed the intestine with 55 ml Coca Cola and 5 ml iohexol (x-ray contrast agent). The hairball passed naturally without further problems.
Why Coca Cola?
Probably because that was what the vets had on hand. Cutting the lower stomach and intestines to remove an object is a major surgical intervention with a lot of risks involved and a long recovery time. If you are so close to solving the problem without having to do that, you try anything reasonable you have at hand.
Cola cannot dissolve hairs, so it certainly didn't make the hairball vanish. However, the acids in Cola can dissolve some other gunk that might have slowed the natural passing of the hairball down.
The introduction of any liquid also inflates the intestine. If the hairball (which is usually an elongated shape) was stuck before a bend in the intestine, this slight inflation could be enough to straighten this section enough to let the obstruction pass through.
Also, the additional liquid lubricates the intestinal wall and the hairball to help it pass naturally.
Maybe some air bubbles from the carbonation can also stick to uneven surfaces and smooth them out, but this is pure speculation. I don't know to which extend the vets removed the carbonation prior to injecting the Cola.
Why iohexol
Simply to be able to observe whether or not the obstruction clears out.
What are the alternatives?
Since I'm not a vet I cannot answer how similar cases are usually treated. The authors of the paper implied that they tried to grab the obstruction with surgical instruments like endoscopic tongs but couldn't reach it. I know that pulling objects and tissue through the endoscope is common practice and can involve cutting or otherwise fragmenting too large objects first.
I have no idea if a "hydraulic" approach is common or something these vets came up with.
Personal opinion
While I suspect that simple water or carbonated water could have been successful as well, I respect the authors Wilson, Savanah; Dobbins, Devin; Kawalilak, Lukas; Parambeth, Joseph C. for trying out their unusual approach. Solving a possibly lethal intestinal obstruction without major surgical intervention is an accomplishment.
That being said, there's a possibility that the obstruction could have cleared on it's own and the vets admitted that more studies need to be done to determine the effectiveness of the procedure.
