Ah, just a matter of finding the right abstraction / realization that allows you to see why each algorithm always finds the right answer. For this one, a simple two-part realization does it:

(----- the full list -------)
       [-- max sum --]
  <neg><pos><- plus ->
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If a subrange has the maximum sum, then splitting that range into two new subranges will always give you two ranges that each have non-negative sums (otherwise the other new subrange would be an even better choice). In particular, if "<pos>" has a negative sum, then "[-- max sum --]" does not have the maximum sum.

Conversely, any non-empty subrange directly to the left ("<neg>") of the max-sum subrange will have a negative sum (else adding that in would give a better sum).