When someone demands his/her pound of flesh, he/she wants something that he/she is entitled to. He/she wants it even though it might create problems for the person it is demanded from.
In other words, he/she ask for what is due to him/her, even though he/she knows it’s going to create problems for the other person.
For example, the family’s distress had absolutely no effect on the greedy moneylender. He was determined to get his pound of flesh.
The expression comes from Shakespeare’s, `The Merchant of Venice’.