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I've worked through Kadane's and love its elegance. I am worried though, unless asked specifically about arrays and it's obvious that's what's for dinner, I won't know to use it.

What are some real-life problems and applications where this algorithm is key? How would one go about recognizing them in the wild?

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See the Wikipedia article on the maximum subarray problem for applications to genomic sequence analysis, computer vision, and data mining.

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    $\begingroup$ I have been intensely doing computer vision for the last 30 years (both R and D), and never met a single use of this. In particular, the 1D problem doesn't fit with images. :-) $\endgroup$ – Yves Daoust Jun 3 '19 at 8:03

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