Roderick says that he shares a special kind of connection with Madeline. What is he talking about exactly?

First, note that Madeline and Roderick are twin brother and sister. Within "The Fall of the House of Usher," they have been living together for an extended period of time, in a state of isolation, with Roderick's mental state deteriorating at the same time that Madeline has been suffering from her illness. Between the isolation and the fact that both inhabitants in the house have been getting steadily worse health-wise, one can assume this factors heavily in the gloomy atmosphere which the narrator finds when he arrives at the house. Madeline's poor health certainly seems to have a role in shaping Roderick's own severe neurosis.
Things become worse when Madeline dies (or when Roderick believes she has died), with the narrator describing Roderick as a broken man, listless and adrift:

His ordinary manner had vanished. His ordinary occupations were neglected or forgotten. He roamed from chamber to chamber with hurried, unequal, and objectless step. The pallor of his countenance had assumed, if possible, a more ghastly hue—but the luminousness of his eye had utterly gone out.

At the end of the story, we learn that Madeline has not been as dead as it had been assumed when she emerges from her coffin during the storm. At the story's end, the two die together, as the narrator flees to watch the house crumble and fall. All three (the house and its two occupants) seem linked together within the story, and all three essentially die together at its end.

In Edgar Allan Poe’s story, the narrator visits Roderick Usher in his home. He learns that Madeline, Roderick’s sister, has recently died. Roderick gets the narrator to help with temporarily interring her body until he can arrange for a proper burial. As they look at her body in the casket, the narrator notices the strong resemblance (“striking similitude”) between brother and sister, although he does not comment on it.
At this point, Usher seems to read his mind and tells him that he and Madeline were twins. They had always shared thoughts that did not need verbal expression, what he calls “sympathies of a scarcely intelligible nature.” Because Madeline has died of an illness of “unusual character,” which the narrator calls “cataleptical” or related to mental disease, by extension Roderick can be implying that he has the same symptoms or fears for his sanity.