I haven't read the study, and refuse to for laziness reasons, but I assume you'd have an understanding of what cognitive decline normally happens, and have a control group that hasn't gotten covid within the timeframe, and then do a between-groups statistical thingy. Once you control for the difference, you can see the effect that covid causes. There is some complicated statistical stuff you do here that I don't remember (and SPSS does it for you).

The sample size is large enough that a decent effect size can be seen.

Of course, that isn't causative proof. For instance, there might be some self-selection bias for people who didn't get covid during the allotted time period. But nonetheless I'd assume that's what they would have done to "prove" a correlation. This stuff gets slammed into you a bunch in undergrad shrinkology.