When we first meet Louisa, she’s Miss Normal – both she and her sister have "all the usual stock of accomplishments" and are "like thousands of other young ladies" (5.45). As we find out more about her specifically, rather than as half of the Musgrove Sisters, she appears to be everything Anne is not: cheerful, high-spirited, and confident. While Anne stays on the sidelines trying not to be noticed, Louisa is happy being the center of attention. Louisa’s appeal to Wentworth is the way her strengths mirror what he thinks of as Anne’s weaknesses, which is an early clue to how much Wentworth still has Anne on the brain.
Wentworth’s constant praise of resolute characters has its effect on Louisa, persuading her that decisively pursuing what she wants is the way to Wentworth’s heart. If you’re thinking, "wait, wasn’t openness to being persuaded something Wentworth saw as a flaw in Anne?" you’re on to something. Perhaps it wasn’t the persuasiveness that was the problem, it was that Anne listened to Lady Russell and not to him.
The novel makes it pretty explicit that Louisa’s fall at Lyme is the result of her refusal to give in to persuasion.
[Captain Wentworth] advised her against it, thought the jar too great; but no, he reasoned and talked in vain, she smiled and said, "I am determined I will:" he put out his hands; she was too precipitate by half a second, she fell on the pavement on the Lower Cobb, and was taken up lifeless! (12.33)
Wentworth’s reasoned advice is placed against Louisa’s willful determination, and determination wins the day…or loses, considering the outcome. Louisa’s refusal to be persuaded by Wentworth, or even wait for him to stop trying to be persuasive and actually be ready catch her, almost kills her, forming some pretty strong evidence for the virtue of being open to persuasion, or at least of looking before you leap.
Louisa’s jump from the wall at Lyme has all sorts of other repercussions in the novel as well. Besides showing Anne’s character to Wentworth in a better light, it also makes Wentworth realize that others have been taking his flirtations seriously and he might have to marry Louisa whether he wants to or not. It’s the concussion heard round the world as Wentworth rethinks his priorities and realizes he’d take Anne over Louisa any day.
But what of the fall’s effect on Louisa herself? Louisa virtually disappears from the narrative after Anne leaves her unconscious at the Harvilles – there’s a steady trickle of information about her recovery, but she never appears directly again. Perhaps that means once she’s no longer Anne’s rival and foil, she doesn’t really matter to the story, or perhaps her transformation into a Byron-quoting bookworm would seem even less convincing if we saw it first-hand. In any case, Louisa’s departure clears the air, allowing Wentworth and Anne to finally get together and have their long-delayed happy ending.