It means different things to different people. The author was born to a Jewish mother and a Christian father. She is Jewish by Jewish law, but attended church as a child and grew up with Christian holidays and traditions.
When people refer to themselves as culturally Jewish, I assume that they are Jewish by birth but don’t attend temple or participate in the religious aspect of Judiasm. I suppose it is like a Christian who doesn’t go to church and celebrates a secular Christmas.
In the article, the author explains it as:
As I got older, things got more complicated. I stopped going to church and stopped believing. I identified ethnically and culturally as Jewish, more so once I moved to New York, the finest place on Earth to be a nonpracticing secular Jew, and married into a Jewish family. I don’t go to temple, but I fry a mean Hanukkah doughnut.