We have to look at truth in two different ways to answer your question, because there are two types of inquiry presented here - explicative and interpretive.
Suffering is necessary in order to come "into the truth" in both these plays, in part, because the truth of the facts are painful to the protagonists of each play. Agamemnon is betrayed and murdered by his wife and Oedipus is told that he has unwittingly murdered his father and married his mother. Discovering the facts for both of these protagonists is clearly a painful enterprise and a cause of suffering.
For each character, to know the truth is to suffer. That is the first answer to the explicative question, "In what way is the chorus correct to suggest that 'we must suffer into truth'?"
You ask another question here, which is an interpretive one, about what truth lies in these plays. This question relates to the artistic merit, the emotional integrity and the themes of each play. Another way to ask this question would be to ask, "What insights into the human condition are offered by these plays?"
This is a much more open question. We might look at each play and see how much courage is required to really admit to painful facts, to past mistakes, and to responsibility. We might also see these plays as indicative of the real challenges of family life, of loyalty, and of forgiveness. The intepretively oriented truth you find in these plays, in the end, is up to you.