You asked us, "Why did ancient Greeks and Romans eat lying down?"
Reclining and dining in ancient Greece started at least as early as the 7th century BCE and was later picked up by the Romans.
To eat lying down, while others served you, was a sign of power and luxury enjoyed by the elite. People further down the social ladder copied the laid-back dining style, if they could afford to.
I mean, who wouldn't want to stretch out while chowing down, but not everyone was so lucky in ancient Greece. You see, women didn't generally get invited to banquets except for rare occasions like wedding feasts and even then they had to sit upright.
It was only in ancient Rome that customs changed, allowing upper-class women to lounge alongside men, and while it sounds sweet, all that lying down and eating can't have been good for the heartburn.