Panel 1: We find Liz screaming that she HATES writing an apology note because she can't DO it owing to the fact that she doesn't know what to say. This is because she's never actually heard anyone in her immediate family make a sincere apology and (since she's Lynn) also thinks that if she does apologize for stuff she did do, she'll have to apologize for stuff she did not do.
Panel 2: Since Elly is all about advising people how to do things she'd never do herself, she asks an angered Liz what's so hard about writing down "Dear Mrs Grunion, I'm sorry, Sincerely, Elizabeth."
This is rather stupid of Elly because if one is to write an apology note, it must not be so general in nature. She should be advising Liz to apologize for giving in to peer pressure and endangering herself and her fellows.
Panel 3: Liz screams that she hates writing an apology because it's the worst punishment she's ever had. This is because it's pretty much the only punishment she's ever had.
Panel 4: Since Elly thinks that the point of raising children is to make them miserable, she thought-bubbles that it's the best punishment she's ever had.
Summary: How would Elly know about punishing her children anyway? If she's not giving them the wrong ice cream or making them hate chores and reading by making those punishments, she's doing what she's doing now and letting someone competent do it.