The reason it looks like magic is because it requires at least one person who Truly Gets It. It's easy to get business people to buy flash, because they cannot measure substance, nor can they differentiate between skill and luck with past experience.
For Agile methods to work, you need at least one person who sees the whole big picture, and given enough time and/or other good people to work with, could implement everything from scratch. Once you have that person and a commitment to make the project work, you'll succeed. If you don't have that person, your odds are just as good as any other poorly managed software project.
I worked on a project where we had 6 months to do more than others had done in 18-24 months in the past. The requirements were subject to change on the fly, but the big picture was understood well enough by everyone to make sure that we kept going down the right path. We bought three off the shelf software components that were supposed to be the building blocks of the project. In the end, we used one, abandoned one, and had to do a lot of work to make the third one usable for our purposes. From a functionality perspective, buying the software covered 10-20% of the work that needed to be done to create a viable solution.
There were several times where the management direction would have badly crippled us further into development. However, our management trusted our recommendations and later in the process learned that what we were advocating had been right all along, but wasn't obvious until core requirements changed later.
Without someone who has a deep understanding of software development and all of the things that go into it, the whole process looks like magic.