Own is often used as an emphasiser to enhance or draw attention to the closeness or possessional aspect of the relationship.
- I'll be staying in my own home tonight. Draws attention to the fact that it's not anyone else's home. Perhaps I'm returning from holiday or have just purchased my first house, and so am excited to finally be staying there.
- Vincent van Gogh cut off his own ear. Cutting off one's own ear is quite unusual. Own makes it explicit that his refers to van Gogh, and not some other man.
In this specific instance, Detroit's own calls attention to the fact that Eminem hails from Detroit. However, in the movie, Eminem is performing in New York City1. As mentioned by Jim, this phrasing is often used when introducing a performer who is appearing in their hometown, but this need not be the case. Own can be used any time you want to add emphasis; for example, to add a bit of verbal flair to one's speech, such as a talk show host may be wont to do.