The meaning of "take a dig at someone" is:
The definition of "take a dig at someone" is:
(verb) to insult someone when other people are already saying negative things
Mom: Son, you are too lazy. You must study harder.
Dad: Yes son, if you don’t study, you won’t get a good job. You also need to clean your room more. You are so messy!
Son: Okay. Don’t take a dig at me! Leave me alone!
Employee 1: Our boss is always late for meetings. That bothers me.
Employee 2: Yeah and the meetings are always too long. He should make them shorter.
Employee 3: And he should stop wearing those ugly ties.
Employee 2: There’s no reason to be taking a dig at the boss.
My ex-girlfriend took a dig at me when she told my new girlfriend that I was lazy and unmotivated. I’ll get my revenge, one day.