This seems like the sort of article which will be quoted, misquoted and pass into mythology.
It's a brave attempt to link nurse staffing levels to one measure of hospital quality - death in hospital within 30 days of surgery. Not too difficult to spot when a patient has died, so the statistics should be pretty accurate here (though there are some well known problems if you only look at in-hospital death - this could be low because moribund patients are shipped out to care homes or other facilities).
The real problem with the outcome is that it only acts as a marker of quality if you can adjust for how ill the patient was on admission. So there are some heavy adjustments for severity of illness, and some fancy techniques (generalised estimating, hierarchical models) to allow for the fact that so many hospitals were studied.
The predictor - nurse staffing per patient - is more difficult to measure. They wisely decided not to rely on official figures and carried out direct surveys of the hospitals. Response rates were low. Frankly I'm surprised they were able to show anything at all.
The actual results are quite striking, as are some of the incidental remarks: in Spain all nurses are graduates, and in Germany none are.