ELI5 Wrinkled Fingers - why water makes our fingers wrinkle?

How did they disprove the common assumption that our skin wrinkles or water aging occurs because water makes it swell? What did they observe and how is this fact used in the medical field today? What evolutionary purpose does wrinkled fingers serve? What other medical conditions could wrinkled fingers be linked to?
 
... we explain like I'm five
Thank you to the r/explainlikeimfive community and in particular the following users whose questions and comments formed the basis of this discussion: forgivethepast, axel927 and dday0123.
Nature article referenced in episode:
https://www.nature.com/articles/nature.2013.12175 
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