Technically, Persuasion is told by an omniscient, third-person narrator who can get into the heads of all the various characters. However, practically speaking, the novel is almost entirely told from the third-person point-of-view of the protagonist, Anne Elliot.
This is important, because every Jane Austen novel functions by manipulating point-of-view. We as readers are privy primarily to the main character's perspective, with other people's thoughts or information rarely brought in. In this novel, therefore, almost all of what we see and hear is mediated through the consciousness of Anne Elliot. What she knows, we know; and what she doesn't know, we don't know. When the narrator offers us background information, such as the history of Anne's relationship with Wentworth, this is information Anne already knows.
This is important because, by seeing the world through Anne's eyes, we are as baffled and surprised as she is by the way events work out. Since Anne, for example, is oblivious to the idea that William Elliot is a scheming manipulator until she hears the story from Mrs. Smith, we are too. We don't, in other words, have the narrator getting inside Mrs. Smith's head to tell us ahead of time that William is untrustworthy. Likewise, we are as surprised as Anne to discover William has a relationship with Mrs. Clay: because Anne doesn't see it coming, neither do we.
By telling the story primarily through a character who doesn't have all the information, Austen can surprise and delight us.