Maybe it means to make someone do something by shaming him/her?
That's close. I'd say it means to make someone do something by making them feel shame if they don't do it.
Let's say I want you to give money to a charity. I may entice you to give ("If you give $50, they'll give you a free coffee mug!"), or I might encourage you to give ("This charity helps a lot of people; you should consider donating!"). I might also badger you to give, if I keep asking you day after day, "Have you given to that charity yet?"
However, if I were to shame you into giving, I would try to make you feel ashamed for NOT giving. For example, I might say:
This charity gave you a lot of help when you were in need. Now that you've become wealthy, you're not going to help them out?
If that compelled you to give to the charity, then one might say that I shamed you into giving.