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Perhaps an obvious question, but why do APIs need to be browsable?

If you are still asking yourself that question, I strongly recommend you read this book: http://restfulwebapis.com/

It's a legitimate question. I understand why URLs need to be browsable.

How does a browsable API work when it may need certain HTTP headers, request signatures, etc?

That's in the contents of the book.

They don't need too, but it sure-as-hell improves accessibility & usability.