Happy to help. This is a strengthen with a necessary premise question, so we're looking for the answer that, if false, makes the argument fall apart. We can use the negation test with each of the answer choices to see which answer is right. If the negation doesn't wreck the argument, we know that answer choice isn't the correct answer.
(E) says, "Happiness is not genuine unless it is based on something that is painful to obtain."
(E)'s negation is, "Happiness can be genuine even if it isn't based on something that is painful to obtain."
The problem with (E) is that the argument already provides us with conditions that have to be met for general happiness. We do not need to add more conditions, and happiness doesn't have to be painful to obtain.
(B) says, "People who are not dissatisfied with themselves are less likely than others to pursue personal excellence."
(B)'s negation is, "People who are not dissatisfied with themselves are more likely than others to pursue personal excellence."
If people aren't dissatisfied with themselves were more likely to pursue personal excellence, then the argument would be wrecked because this would invalidate the entire reason for someone not to accept themselves as they are. Thus, (B) is required in order for the argument to stand a chance, so it's the correct answer, as it's a necessary premise.
Does this make sense? Let us know if you have any more questions!