"Outcome" in medical research is also sometimes called an "endpoint" and in a statistics class you might have heard it called the "dependent variable". It's the thing you measure to find out whether the thing you're studying is having an effect. An outcome should be operationally defined, so if you want to know whether yellow jelly beans cause cancer, your outcome might be "cancer incidence within 1 year of eating a yellow jelly bean".
The study you link to is a systematic meta-analysis. Meta-analysis is a family of statistical techniques to combine research results from multiple studies. Every study makes some estimates of the effect being measured, and these effects are never absolutely precise, they have some uncertainty around them, sometimes expressed with a confidence interval/credible interval. That doesn't mean that studies are wrong if different studies find a slightly different answer, it just means you need a way to decide what the best estimate is if you have a bunch of different estimates. Meta-analysis is the way you do this quantitatively.
When you do a meta-analysis, you have to decide what you're looking for. That means identifying both exposures and also outcomes that are relevant to your study; also other aspects like who is the studied population (adults? kids? receiving care in France? tea-drinkers?). For example, do you want to do your meta-analysis about all jelly beans, or just the yellow ones? Do you want to know if yellow jelly beans cause cancer, or heart disease, or tooth decay? Or do you want to know if they cause increased mortality by any cause? Whatever your goals are, you have to define them somehow so you know what studies to include. You can't really combine the estimates of cancer incidence with the incidence of tooth decay, or you'll get a mashup that's not interpretable - how is an estimate of the incidence of "cancer or tooth decay" at all more useful than knowing the two separately?
For your specific study, they identified a range of outcomes they were interested in:
measures of adiposity (e.g. body weight, body mass index [BMI], overweight/obesity, fat and
lean mass);
type 2 diabetes and pre-diabetes (incidence and intermediate markers of glycaemic control);
cardiovascular diseases (incidence and intermediate markers, such as blood pressure and
lipids);
cancer;
dental caries;
chronic kidney disease;
eating behaviour (e.g. appetite, satiety, energy intake);
sweet preference (e.g. subjective measures, sugars intake);
neurocognition;
mood and behaviour; and
asthma and allergies (for children only).
Okay, forget everything I said, they literally have both cancer and tooth decay (dental caries) on the list! Or at least, they're interested in both, but not because they're going to combine them, they're just looking at a lot of different things at once, doing a bunch of separate meta-analyses together. So they're going to find all the studies that measured adiposity (e.g., BMI) and combine those estimates together, all the studies that measured diabetes and combine those together, all the studies that measured cancer and combine those together, etc.
However, they're not looking at everything. Specifically, "variation in faecal microbiota composition" isn't on their list. So, when they went to collect a bunch of useful studies to address their wide goal of estimating everything from cancer to dental caries, they still found some studies that were out of scope with their original analysis plans, and they had to set those studies aside because the study didn't have an outcome that was on their list of outcomes of interest: that study has no outcome of interest.
