Let’s assume you’ve just arrived in a city and don’t really know your way around yet. In this case sometimes a friend or acquaintance of yours might come up, put his or her arm around your shoulders and tell you: “Ich nehme dich schon unter meine Fittiche.”
With that, your friend is jokingly saying that they will look after you. The literal English translation of that sentence is “I’ll take you under my wing.”
Fittiche is an interesting case of a word that is so old that almost nobody uses it as it's original meaning anymore. In colloquial talk, however, it is still widely popular. According to the Duden, it comes from the Mid High German word vitich/vetach and the Old High German fettāh/feddāh, which means wing.
The notion of this phrase comes from the birds’ habit of protecting their chicks by spreading their wings over them.
So, if your friend is offering you to take you under their Fittiche, don't be alarmed it’s just a warm way of saying that they will look after you, offer some guidance and make sure you're on the right tracks.
Sie kannte sich nicht aus, also habe ich sie unter meine Fittiche genommen.
She didn’t know her way around, so I took her under my wing.
Er war noch sehr jung, als ich ihn unter meine Fittiche nahm und ausbildete.
He was very young so I took him under my wing and trained him.