If you've been watching Game of Thrones, there's no way you missed how Littlefinger looked at Arya after she totally wrecked Brienne in hand-to-hand combat. Here's a video - the Littlefinger creepy stare is at the end.
We know how Littlefinger operates. Everyone's a threat, everyone's an ally - when he sees Arya strut her stuff as one of the most dangerous people in Westeros, he realizes there's a big player in the Game of Thrones he wasn't even aware of (a player of single-handedly destroying entire houses if she wishes). Arya is already suspicious of Littlefinger's motives and his influence on Sansa, who she can tell has higher aspirations than "good sister" to Jon in the North. Oh the tangled web we weave! Arya follows Littlefinger around and, seemingly unnoticed, observes him receiving a secret scroll which he locks away in his room. She sneaks in a reads it! Though the scroll is on screen for only a couple seconds, some dedicated fans paused the show to read the note's contents and interpret its meaning.
Robb, I write to you with a heavy heart. Our good king Robert is dead, killed from wounds he took in a boar hunt. Father has been charged with treason. He conspired with Robert’s brothers against my beloved Joffrey and tried to steal his throne. The Lannisters are treating me very well and provide me with every comfort. I beg you: come to King’s Landing, swear fealty to King Joffrey and prevent any strife between the great houses of Lannister and Stark. -Your Faithful Sister, Sansa
But oh my, we find out just as she's leaving Littlefinger's room that Baelish himself planted the note for Arya to find! Littlefinger, you devilish mastermind! Arya, not knowing that Sansa was forced to write that note under threat of death by Cersei, and already doubting Sansa's faithfulness to Jon's cause, may see the note as a major betrayal and cease to trust Sansa. Littlefinger is attempting to drive a wedge between the Lady of Winterfell and Westeros's most accomplished rogue assassin!
Arya is very used to being more clever and more stealthy and smarter than any of the people she's up against, and she hasn't dealt with Littlefinger for awhile. So 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 that bond is the more definitively he is caught on the outside of it.
Don't get me wrong, Littlefinger, your plan is a good one. In any other case, my money would be on its success...but you don't know the Stark children as well as you think you do. You may be able to manipulate power and perspectives, but if you wind up on Arya's list there's nothing in the Seven Kingdoms that can save you from your fate.