The seemingly modest article “the” actually leads an exciting life. Imagine if you killed the “at” in front of “eating at the table”: you’d be witness to an act of xylophagy!
Putting “the” in front of a word specifies the object. You’re not just talking about any table here. We’re talking about THE table. (Figaro once stayed at a hotel called “The Hotel” in Vegas. He pronounced it “Thuh Hotel” at the front desk and they almost threw him out.)
Eating “at table,” on the other hand, has a snobbish, Anglophile feel to it. It implies an un-American style of eating with a knife and fork. One certainly does not eat one’s McDonald’s Happy Meal “at table,” does one?
So next time you’re at table, pull up a “the” before chowing down.