This article and title seem fairly clunky because they are using a single aspect of obesity to make larger inferences. I don't necessarily disagree with their conclusion, just that the way it is written in this article doesn't lay it out very straight forwardly. Here are the basic findings in the article.

They took the medical understanding that obesity is often accompanied by inflammation (they focused on a specific form of inflammation) and they did some tests on mice showing that gaining a lot of weight may possibly trigger epigenetic changes which result in the inflammation persisting even after the weight was lost.

The conclusion is that losing weight (for overweight mice, and possibly then people) comes with a host of health benefits but that some of the bodily changes caused by being overweight (specifically inflammation) may be longer lasting and may not disappear just by losing the weight again. In those cases, they believe we may need to consider the longer term effects of being overweight even AFTER the weight is lost and not simply stop examining the effects of obesity once the weight is lost.

Even more simply: Obesity causes health problems. Some of those problems may linger even after the mouse (or person presumably) is no longer obese. Some obesity caused bodily stress may persist even after weight is lost and should continue to be taken into consideration.