You can read about it here. https://www.delta.com/us/en/delta-sky-club/access
There's three main ways to qualify - travel on a premium ticket, have the Delta American Express Reserve Card, or purchase an annual membership. Other less common ways are if you have the Amex Platinum or Centurion you also have access, same if you're a SkyTeam Elite. Access can't be purchased for one-time use unless with a qualifying guest. Not to be overly glib, but Delta's position on lounge access is basically "if you have it, you'll know." (You can also access the lounge for $29 with the Delta Gold and Platinum American Express cards, but only until January 30, 2020)
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