This passage refers to Fanny's relationship with her younger brother, William Price. The first line,"Children of the same family, the same blood, with the same first associations and habits, have some means of enjoyment in their power, which no subsequent connexions can supply" refers to siblings. Austen is arguing that siblings who have been raised together enjoy each other's company in ways that are different from how they enjoy their other friends. She is arguing that the relationship between siblings is even stronger than the relationship between friends.

She goes on further to state that "it must be by a long and unnatural estrangement, by a divorce which no subsequent connexion can justify, if such precious remains of the earliest attachments are ever entirely outlived." In other words, siblings will keep their affection for each other, unless they have been separated from one another. She also says that more often than not, siblings are separated and they loose their affection for each other.

When she says, "Fraternal love, sometimes almost everything, is at others worse than nothing," she is referring to siblings who have lost their affection for each other, due to separation or some other reason. She is arguing that very frequently, siblings loose their love for each other, and while their relationship would normally be strong, it becomes even worse than indifference--perhaps even hatred.

Finally she states that William and Fanny price still have a very strong relationship and a very devoted affection. Their relationship has not been separated by selfishness ("opposition of interest") or jealousy ("cooled by separate attachment"). Time and distance from each other have only made their relationship stronger.