This sentence doesn't make any sense. They should have at least published a 95% confidence interval in the difference or ratio between time to recovery.
If your significance level is .05 (meaning you are willing to accept a 5% chance of a type 1 error in which you reject the null hypothesis incorrectly), you can use either a confidence interval or a p value. You could look at a 95% condfidence interval and see if the % faster time to recovery range includes the null hypothesis of zero, or you could calculate a P value to see if it is less than .05. Both of them will give you the same conclusion.
Based on the P values given, you know that the 95% confidence interval for % faster time to recovery will not contain 0% (nor will the 99% interval or the 99.9% interval). But, the % difference in mortality at a 95% confidence interval will contain zero, since p>.05.
"Preliminary results indicate that patients who received remdesivir had a 31% faster time to recovery than those who received placebo (p<0.001). Specifically, the median time to recovery was 11 days for patients treated with remdesivir compared with 15 days for those who received placebo. Results also suggested a survival benefit, with a mortality rate of 8.0% for the group receiving remdesivir versus 11.6% for the placebo group (p=0.059)."