There is not likely to be a single mechanism involved, but the study authors suggest that high chloride is the main issue; they refer to previous work showing a whole variety of issues that can be caused by higher-than-normal chloride levels.
Although saline contains roughly the same sodium content as blood plasma, it contains a lot more chloride than usual, because blood plasma contains several other anions such as bicarbonate.
However, please see the rest of my answer: this study shows a fairly modest effect and is certainly worth some more investigation but might not be as definitive as the news makes it seem.
The study in question
I assume the news article you reference refers to this recent paper:
Semler, M. W., Self, W. H., Wanderer, J. P., Ehrenfeld, J. M., Wang, L., Byrne, D. W., ... & Guillamondegui, O. D. (2018). Balanced crystalloids versus saline in critically ill adults. New England Journal of Medicine, 378(9), 829-839.
I am constantly befuddled by why journalism considers it appropriate to refer to a study with no citation at all of the original article, only barely noting one of the original authors and only after mentioning an interview with someone completely different. Even their link to "New England Journal of Medicine" is not a link to the journal but to other articles written by that news organization that also refer to NEJM.
Careful interpretation of the study results
First, note that this study is talking about a fairly small difference. It isn't like saline is causing people to just drop dead, and you recognize this in your question, writing:
Presumably the traditional saline solution does not contain a "wrong" or dangerous concentration of salt
Rather, the study showed that 1139/7924 (14.3%) with "balanced fluids" had an adverse kidney event versus 1211/7860 (15.4%) with saline.
They also reference two previous papers, one from the same authors:
Young P, Bailey M, Beasley R, et al. Effect of a buffered crystalloid solution vs saline on acute kidney injury among patients in the intensive care unit: the SPLIT randomized clinical trial. JAMA 2015;314:1701-1710.
Semler MW, Wanderer JP, Ehrenfeld JM, et al. Balanced crystalloids versus saline in the intensive care unit: the SALT randomized trial. Am J Respir Crit Care Med 2017;195:1362-1372.
each of which showed no difference between "balanced fluids" and saline.
Therefore, I think concluding
"...that it's now conclusive that a balanced combination of ingredients beyond NaCl is safer."
is not necessarily true. In Young, et al., 102/1067 (9.6%) versus 94/1025 (9.2%) of patients showed signs of acute kidney injury for the "balanced" versus saline groups - a difference that was not significant (i.e., there are not enough subjects in the study to know that an effect this size is unlikely due to chance), but numerically the "balanced" group was worse. In Semler, et a. 2017, major adverse kidney events were 128/520 (24.6%) "balanced" versus 112/454 (24.7%) saline - literally as close to "no difference" as possible with this number of subjects (i.e., 111/454=24.4%).
Therefore, although this latest work shows a significant result with a larger population size, if you consider the previous work in this area the results are definitely mixed. The more recent Semler et al. 2018 paper's result is based on a p-value of 0.04, which is just below a commonly used threshold of 0.05 for deciding whether a result is significant. That doesn't mean the result is wrong, but it means that it should be considered in the context of other work being done in the area.
This study is also focused specifically on kidney issues, and although they have some other general secondary measures like deaths, the study is not designed or able to tell whether certain patients may actually be harmed by use of 'balanced' IV rather than saline IV.
To your actual question,
How might IV-saline cause kidney damage that seems to be less likely with “balanced fluids” IVs instead?:
The authors refer to other work showing that high chloride levels lead to a variety of issues; saline has a higher chloride level than normal blood plasma levels, so they suggest that if there is an issue with saline, the chloride levels are a likely cause, but this study has no ability to test those mechanisms.
Relevant quotes from the article, with emphasis mine:
In preclinical models, the high chloride content of saline has been reported to cause hyperchloremia, acidosis, inflammation, renal vasoconstriction, acute kidney injury, hypotension, and death. Studies involving healthy volunteers suggest saline may decrease renal perfusion through chloride-mediated renal vasoconstriction.