I think others have specified it better than I can, but essentially, when done well, achievements are able to help a user either 1) complete parts of a game that they wouldn’t otherwise or 2) complete the game in ways they wouldn’t otherwise.
A Broad Semester does this with a photo library. Because there are a number of branching paths, it is easy to miss out on some branches without realizing it. With a photo library, it is easy to know at a glance whether you have hit all of the branching paths or if there is content you have missed.