Nora has many revelations at the moment she says, "Yes, I am beginning to understand thoroughly" (I).
One of her revelations is that Torvald is neither the man she thought he was nor does he truly love her as she thought he did. Earlier in the play, when Torvald becomes annoyed with Nora for fretting that Krogstad will slander Torvald in the press should he fire Krogstad, she is led to believe that Torvald would sacrifice anything for Nora's sake and do anything possible to protect her. We see Torvald saying what looks like a promise to sacrifice himself and protect her in his lines:
Come what will, you may be sure I shall have both courage and strength if they be needed. You will see I am man enough to take everything upon myself. (II)
We see that even Nora herself has interpreted his lines as a promise of self-sacrifice when she says "[In a horror stricken voice]. What do you mean by that?" (II). We also see Nora's determination to prevent her husband from sacrificing himself for her sake when she further says, "You will never have to do that" (II). However, once Torvald has learned that Krogstad has the means to slander his wife's reputation as well as his own, he is only furious. He calls her a "thoughtless woman" and asks "do you understand what you have done?" after Nora proclaims, "Let me go. You shall not suffer for my sake" (III). As a result of his fury, Nora now realizes that all his talk of self-sacrifice and protection was just that--all talk. Furthermore, Torvald does not truly love her; instead, he thinks she is a ridiculous and thoughtless woman.
Another revelation she has is that all these years he has treated her as a plaything because in reality he has never taken her seriously, but instead, thought of her as a ridiculous woman. He has enjoyed her beauty, just as he would enjoy a doll's beauty, but has never taken her seriously, which is why she declares that she has been his "doll-wife" (III).
Hence, the reason that Nora says "I am beginning to understand thoroughly," is because she has realized many enlightening things about her husband and how he has treated her.