From the authors, my emphasis

>Epigenetic alteration is known to exert long-term effects on gene expression and phenotypes37,38. Given the increasingly realized high incidence of post-acute SARS-CoV-2 sequelae (long COVID39), understanding the viral impacts on host chromatin and epigenome will not only provide new strategies to fight SARS-CoV-2 in the acute phase, but also pave the way for unravelling the molecular basis of long COVID for its intervention.

Lots of viruses make use of the host's DNA for their own purposes. This study looked at how SARS-CoV-2 affected host DNA, on a large scale. [more in-depth: DNA is a giant, giant molecule. It's so big that its overall organization itself is really important and affects all kinds of things. A very simple example is that expression of a certain gene may rely on activation of a region on the DNA that is, in 1-dimension, extremely far away. But just like you can loop a rope back on itself to bring two points close together, these points can be extremely far in 1 dimension (along the DNA chain) but right next to each other in 3 dimensions.] This study found that SARS-CoV-2 does indeed modify infected cells' DNA on large scales, and this is important not just for acute infection but can cause lasting impacts after the infection is cleared.

ELI5: The virus bends the DNA around so the cell can't fight the virus as effectively