What special knowledge does Hester feel the scarlet leter gives her?

One of the most important things that the letter gives Hester is sympathy.  She cares for the poor and gives of her meager income to help them --- this despite the fact that often do not reciprocate and some even berate her.

We may speak further of it hereafter. Except for that small expenditure in the decoration of her infant, Hester bestowed all her superfluous means in charity, on wretches less miserable than herself, and who not unfrequently insulted the hand that fed them. Much of the time, which she might readily have applied to the better efforts of her art, she employed in making coarse garments for the poor. It is probable that there was an idea of penance in this mode of occupation, and that she offered up a real sacrifice of enjoyment, indevoting so many hours to such rude handiwork.

It is one of the ironies that Hawthorne explores in many stories:  sin (or the knowledge of sin, real or suspected) can destroy you (Young Goodman Brown, Roger Chillingworth) or it can make you a more sympathetic, better person (Hester, Dimmesdale (as a minister) and Parson Hooper (The Minister's Black Veil)).  More of Hawthorne's ambiguity!