COVID is airborne. It can move through the air, remain in spaces for a long time, and generally transmit much easier through the air than most realize. People don't need to be in real close proximity to the deer to get infected, just close enough to share some air. With lots of people, lots of deer, and lots of COVID, there's lots of opportunities for cross infection.
They compared the genetic sequences of the collected deer samples to available human ones and found 3 that were nearly identical, which suggests direct spread. From the article:
Lastly, the genetic data suggested three spillovers of deer-adapted SARS-CoV-2 viruses back to humans—two in North Carolina and one in Massachusetts. For this, the researchers compared the deer SARS-CoV-2 sequences to sequences of viruses isolated from humans, available via public databases. In all three possible spillover cases, SARS-CoV-2 isolates from human cases were over 99.9 percent identical to deer-adapted viruses collected from deer in the corresponding states.