He loved Catelyn
Littlefinger loved Catelyn so much but he lost to Ned who married her. Littlefinger couldn’t get her out of his mind and as a result he might have felt like helping Sansa because she is her daughter.
She reminds him of Catelyn
According to one theory, Littlefinger might have loved Sansa because she reminds him of Catelyn.
He wants the Iron Throne
Baelish was very clear about his intentions of wanting to sit on the Iron Throne. It is possible that he is interfering in the power struggles taking place in the realm to increase his chances of sitting on the Iron Throne.
To get the support of the north
Baelish could be helping Sansa in order to get the support of the north when he claims the Iron Throne. His plan could also be to marry her to make sure the North will support him.
He actually loves her
Baelish could have fallen in love with Sansa. And while his true intentions aren’t known, he still might have had true emotions for her. (See Game of Thrones: Why Cersei hates Tyrion?)
To get rid of Ramsay
Baelish could have realized that Ramsay is a dangerous and ambitious person and that he must get rid of him as soon as possible. He could have brought that army to help Sansa just to get rid of Ramsay.
Using her to gain power
According to one theory, Littlefinger is just using Sansa to gain power and manipulate people. He married her to Ramsay to gain his support then later he got rid of him when the chance was there. He also used Sansa to gain the trust of the Lords of the Vale. (See Game of Thrones: How Olenna Killed Joffrey)
Why does Littlefinger help Sansa?
Why does Petyr Baelish help Sansa? Why did Littlefinger help Sansa?