On “Game of Thrones: Inside the Episode,” showrunner D.B. Weiss explained the motivation behind Littlefinger’s desire to drive a wedge between Arya and Sansa.
“Arya is very used to being more clever and more stealthy, and smarter than any of the people she’s up against, but she hasn’t dealt with Littlefinger for a while. She gets roped into spying on somebody who's actually leading her by the nose to something that he wants her to have.
He’s looking for a way to prevent this sister bond from developing further because the tighter they are the more definitively he is caught on the outside of it. On the Sansa end of the relationship he’s seen the opening and now all he needs to do is give Arya a cause to display real rage towards Sansa. And, he knows full well that with what’s going on in Sansa’s head with regards to Arya and how dangerous she is, when Arya starts to act dangerous, and act angry, he knows where Sansa is going to turn.”
By actively working to keep the Stark sisters from forming a tight-knit bond, Littlefinger can remain Sansa's confidant. And thus Littlefinger is one step closer to being Sansa's sole counsel, and just a little more powerful.