Arrangements are considered derivative works, which means that they are based off another, original work. This also applies to new editions of books or scores, translations, and movie adaptations. This Guide made by the U.S. Copyright Office says that in order for an arrangement to be copyrightable, "a derivative work must incorporate some or all of a preexisting “work” and add new original copyrightable authorship to that work."
As an arranger, the only parts that are copyrightable are the additions you make to the original work. So you don't own the original parts of the music, but the way you arranged the piece and any/all changes you made to it are owned by you. You can still register a copyright on an arrangements, but only on the parts you created.
You do still have to be licensed to make a derivative work. Authors of copyrighted material have the exclusive right to authorize and create derivatives, so if you want to arrange a work, you need to get permission from the original creator first. This can usually be done by contacting the composer or whoever manages their estate. You may be asked to sign a license and pay a fee and will likely need to pay royalties based on your agreement with the original composer.