"A breakthrough infection is a case of illness in which a vaccinated individual becomes sick from the same illness that the vaccine is meant to prevent. Simply, they occur when vaccines fail to provide immunity against the pathogen they are designed to target."

via https://en.wikipedia.org/wiki/Breakthrough_infection

As far as "breakthrough infection increased risk of death", that's explained a little further down on the page in the article linked here on HN:

  Risk of death after SARS-CoV-2 infection was highest in unvaccinated Veterans regardless of age and comorbidity (Fig. 3). However, breakthrough infections were not benign, as shown by the higher risk of death in fully vaccinated Veterans who became infected compared to vaccinated Veterans who remained infection-free.

This is a pretty long-winded way of saying that some people who were vaccinated still became infected, and some of those infected people did die.

The importance of a study like this is to understand how the efficacy of the vaccine holds up over time, which can help inform public health recommendations and policy-making decisions.

"Breakthrough infection" means tested positive for COVID despite vaccination.

However, it's important to note that "breakthrough infection" is different from "breakthrough disease". It is possible, and likely common with the delta variant, to be a carrier with zero symptoms.

A breakthrough infection is just a COVID case in someone who's been vaccinated. In that particular sentence they're just saying that people who developed a breakthrough infection did have a nonzero risk of death, but the vaccines were still fairly protective.