The feature is a middle ground between leaving your embarrassing photos for everyone to see and permanently deleting them. It allows you to hide, or "archive," photos that you don't want on public display. It won't remove them from the service (and will retain all likes and comments), but only you will be able to access them. You also can restore archived photos back to your public profile, allowing you to change your mind (or hide certain photos during, say, job interviews).
Instagram is hoping that this feature will change the way its users interact with the service. According to The Washington Post, teenagers routinely delete the bulk of their pictures off Instagram, keeping their profile at 25 pictures or fewer. Presumably, the service hopes these users will archive, rather than delete, their pictures, so that they remain in Instagram for possible restoration at a later date.