Whence exotic spheres?

I have tried to wrap my mind around differential topology several times over the past few years but never been able to do it well. And I am really confused now that I have read this article. My understanding was that DeRahm's theorem meant that any continuous morph on a smooth shape could be replaced by a smooth morph that does the same thing. But here we have spheres where that doesn't work.

Is there an intuitive explanation of how it really works (or a technical reference I can look up)?

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