By What Mechanism can Felines Reverse Diabetes?
I spent a couple of years working in a diabetes research institute.  And you did hear right. 

Cats dogs monkeys all can be susceptible to diabetes, although rats and mice don't seem to have diabetes that behaves the same way as humans - I don't think a mouse has ever been observed to get diabetes without specific gene knockouts.  One could argue that mice simply do not get diabetes the way people do. 

There are two causes of type 2 diabetes in the research world.  The first happens because the animal may have a genetic susceptibility and environmental conditions induce the disease (like most people).  The second happens because of mutations in one specific gene or another pretty much cause it to happen when the animal gets to a mature age (google term: mature onset diabetes in the young (MODY)).  MODY genes have not proven to be a great model for type 2 diabetes treatment in most people (about 1% of diabetics have a MODY diabetes though). 

Cats and dogs have been animals of interest as models of diabetes because their symptoms are human like and have shorter reproductive cycles.  Some cats (like Burmese) clearly have a greater genetic predisposition to diabetes, but cats who are obese, have high glycemic diets, or who get little or no physical activity are more likely to develop diabetes. 

As with humans, the condition leads to the insulin secreting cells of of the pancreas to slowly degrade and if these cells are gone, there is no recovery from type 2 diabetes. All this given its likely that cats get diabetes that is similar to the human condition. 

In treating cats its been shown that such diabetes can be reversed.  The most important thing to do - common to all treatments - is to change the cat's diet - restrict calories if the cat is obese and change high glycemic food.  Additionally, insulin or drug treatments that reduce insulin resistance (such as are given to humans) can restart the islet cells that secrete insulin if the animal is lucky and don't always have to be lifetime prescriptions. 

The reference I have does not give remission statistics, but it must work a reasonable amount of the time.

But I wouldn't want to leave you with the impression that cats are special.  It turns out that human diabetes type 2 is often reversible.  This reference estimates 80% of the time. This study mostly focuses on the same treatments - change of diet and some fasting, but no exercise regimin as the cats. 

There's probably nothing magical about cat diabetes 2... ;)