From the Key Points:
CAUTION: The results from the present descriptive study are unable to justify a causal relationship between body mass index and injury location distributions. Therefore, it remains open to speculation why different runners are more likely to sustain an injury in certain parts of their bodies.
I think the results of this study are great, while the title might be misunderstood to mean it is safer.
My main takeaway is that being overweight and obese were not more likely to correlated with injuries. That is great because it means that there is not a reason to discourage running based only on being overweight or obese due to a fear of injury. Even if the results are skewed for a variety of reasons such as running frequency, duration, speed, running style, the overall outcome is that running does appear to be as safe for those with higher BMI as compared to normal-weight.