Why is Marble Hill part of the borough of Manhattan?
Because originally the Harlem River's path looped up to that bit of the Bronx. Check it out at the second related link.
So in 1895, a deep, wide trench was completed (called the Harlem River Ship Channel), cutting Marble Hill off of Manhattan and making it into an island. In 1914, the narrow, dangerous stretch of river that separated Marble Hill from the Bronx was filled in, making Marble Hill geographically part of the Bronx. However, to this day, the Marble Hill neighborhood is still officially a part of Manhattan, even though it's in the Bronx now.
If you're interested in learning more about the history of the Marble Hill area, I have included a link to the Wikipedia page about the neighborhood.