The most reasonable explanation for Littlefinger's actions is that he gave Sansa to the Boltons in order to do 2 things:
1. Reveal Sansa's survival to the world without Littlefinger taking the blame for having rescued her from Cersei.
2. Fracture the Bolton-Lannister alliance.
In doing so he would be able to send Sansa to the North and use Lannister men and gold to take the North. Attack the North, somehow making it past Moat Cailin (he may have inside agents who could undermine the Moat Cailin garrison, we do not know the full extent of Littlefinger's network), attack the Boltons while they're expecting Stannis from the North, kill whichever warlord is left standing after the battle of Winterfell and assume control.
This plan would make it clear that Littlefinger does not love Sansa, as it discards Sansa and potentially causes her death for a strategic move that would not at all require sending Sansa to the Boltons. Considering how he started the war of the five kings, breaking alliances is not something that comes difficult to him. It is likewise possible that he could've simply marched north on the pretext of reinforcing the Boltons against Stannis, wiping out whichever force survives when he arrives. He also provides no explanation to Sansa as to why he sent her to Ramsay, and when questioned on whether he knew Ramsay was the little shit he was, Littlefinger tells Sansa that he didn't know - despite his job profession being a spylord who knows who everyone is and what their vices are. His only love is for controlling the Iron Throne, this would be a change from before wherein Littlefinger wanted to control the person in the Throne instead of sitting on the Throne himself. Yet if this was the case, he would have seized the North for himself, as he has two mandates to seizing the North (he has one from Cersei, he could also seize it for Sansa and deprive her of any real power, ruling the North just as he rules the Vale by proxy), and is in command of an army which has no realistic opposition in the North. Yet has delegated his own army to Sansa, elected Jon Snow as King in the North despite having no control over Jon (and indeed, Jon being hostile to Littlefinger) and has not ostensibly told his Vale men what the hell they're achieving in the North - having received no reward to be there as of yet.
Most importantly, the Boltons in the books marry a Stark - one which is actually a fake. The Boltons do this because ostensibly, no one besides Jon Snow is alive who can tell that the Stark they have is a fake, because Rickon and Bran are presumed dead, Ned and Catlyn are dead, Arya presumed dead, Sansa missing and Rob dead. Oh, and I suppose Theon could tell who was a fake, but as Theon was under their control that only made their case stronger - they could use Theon to prove their fake Stark was real. The book Boltons do this to strengthen their claim as the Northlords are understandably upset that their act of treason and ungodly breach of guestright has led to their mastery, and so the Boltons need the legitimacy of a Stark marriage to rally the Northerners around them. The show writers evidently felt this plot point was important, but introducing fake Stark would be too distracting or too complicated for the audience to deal with. Thus they decided to write Sansa into the fake Stark storyline, with the obvious consequences that followed.