Nowhere is the truth of this statement more clearly shown than in Chapter Twelve of this excellent novel, which is when Dimmesdale is joined on the scaffold at night by Pearl and Hester. The way in which Pearl is, in the words of your quote, a "medium" that allows them to express their true identity is made perfectly clear by what occurs when Pearl and Hester step up to the scaffold and Pearl holds hands with Dimmesdale, her father, whilst already holding on to the hand of her mother. Note how the text describes what happens:
She silently ascended the steps, and stood on the platform, holding little Pearl by the hand. The minister felt for the child's other hand, and took it. The moment that he did so, there came what seemed a tumultuous rush of new life, other life than his own, pouring like a torrent into his heart and hurrying through all of his veins, as if the mother and the child were communicating their vital warmth to his half-torpid system. The three formed an electric chain.
Note how the contact unites the three of them and restores life to the ailing and sickening Arthur Dimmesdale. There is something revitalising about the contact that he has with Pearl and Hester, with Pearl acting as the medium between them, that reinvigorates him and gives him "new life" when he had been atrophying in his guilt.
A life-giving symbol, little Pearl represents the bondage of sin and love that exists with Hester and Dimmesdale. As such, she does not reach full humanity until Chapter XXII of Nathaniel Hawthorne's seminal novel. In this chapter, Pearl acts as an extension of Hester Prynne and Arthur Dimmesdale by connecting the physical and spiritual triangle; with this act she, thus, provides redemption for her father. Her act, symbolized by the kiss, causes Pearl herself to be "freed" as "a spell was broken" and she becomes fully human since her "errand as a messenger of anguish was fulfilled."
After Pearl kisses the clergyman, Arthur Dimmesdale is able to make his redeeming confession as he "lookest far into eternity." He praises God,
"He hath proved His mercy...By bringing me hither to die this death of triumphant ignominy before the people! Had either of these agonies been wanting, I had been lost for ever! Praised be His name! His will be done! Farewell!"
For Hester, it has been the wearing of the scarlet letter, of which Pearl is the representative, that has effected her redemption. In her charitable deeds of helping the sick and sitting with the aged and offering them comfort Hester performs penitential acts that redeem her and exhibit her kind nature. Without the letter upon her breast, ironically, Hester may well have not been accepted into homes of illness and death so readily and been afforded the opportunities to redeem herself through good works.
In Chapter VIII when Hester is before the Governor and questioned about her fitness as a mother, Hester explains how Pearl is the expression of herself,
Alone in the world, cast off by it, and with this sole treasure to keep her heart alive, she felt that she possessed indefeasible rights against the world, and was ready to defend them to the death.
“God gave me the child!” cried she. “He gave her, in requital of all things else, which ye had taken from me. She is my happiness!—she is my torture, none the less! Pearl keeps me here in life! Pearl punishes me too! See ye not, she is the scarlet letter, only capable of being loved, and so endowed with a million-fold the power of retribution for my sin? Ye shall not take her! I will die first!”
As the link between them and the reminder of their sin, Pearl acts as a catalyst for redemption in both Hester and Arthur Dimmesdale. Indeed, she acts as the liasion between the earthly and the spiritual for her parents.