"To take a dig at" means to make sarcastic or critical remarks to someone, often while keeping a disarming smile on your face.
"Is that a new car? I didn't know that Public Assistance paid that well."
"That dress is perfect for you - the neckline draws attention away from your non-existant bosom."
"That's a lovely maternity blouse. Oh, you're not.....?"
"Where's the beef?" was a phrase that originated in a TV commercial for Wendy's, the fast food chain. YouTube - Wendy's (Where's The Beef)
Eventually it became something of a slang phrase used to question the substance of an idea or product. For example, during the 1984 presidential primaries, Walter Mondale asked opponent Gary Hart, "Where's the beef?" when Hart continually promised "new ideas." However, the phrase is rarely used today, other than for nostalgia value.