First off, it seems like a bunch of siblings are arguing or fighting in front of their mother.
"환갑" basically means "60 years old", and "제사" is a traditional Korean ritual, remembering, honoring and showing respect for ancestors who are now passed away. "상" is just a table where food and meals are served. The first two questions are basically a couple of sarcastic rhetorical questions that are criticizing the siblings' impolite, childish behavior. "환갑 돼서도 싸울래? 내 제사상 앞에서도 싸울래?" literally means "are you going to fight each other even when you are 60? Are you going to fight in front of my "Jesa" table?" Those two sentences more appropriately translated for those who are not very acquainted with Korean culture could be, "are you going to behave and fight each other like this even when you become 60? Are you going to argue like this in front of my deathbed?"
The following two sentences are pretty much straightforward. "I am your mother and the elder of this family. How could you behave so insolently/impolitely/rudely in front of your mother (your parents)?"
In the next two sentences, the speaker is lamenting her situation, her children's lack of respect and care for her. "Not one of you bothered to ask me my opinion (on this matter). None of you care about/give a damn about/worry about how I am living all by myself." "헛키웠다, 헛키웠다," means "I've raised you wrong".
며느리 means "daughter-in-law", and the last sentence means "I am embarrased that my daughter-in-law had to witness this insolent behavior". It could also be interpreted as the following: "(because of you and your misconduct) I am ashamed to even look my daugher-in-law in the eye." The sentence that came before and this last one demonstrate not only her sorrow and disappointment but the shame she feels as a mother who feel humiliated by her own sons' misconduct.