My way of understanding the solution:
We modify the definition of one operation: rather than incrementing n - 1 elements, we say one operation is:
incrementing n - 1 elements, then decrementing all n elements;
It is intuitive that such a modification would not affect the final solution in any way, since it's the difference that we care about;
And also by the spirit of this definition, the equivalency between incrementing n - 1 and decrementing 1 is much clearer.
To equalize all elements with minimum number of moves, we have to decrement all elements to be equal to the min value, because the only legal operation available is decrementing, and decrementing all values to a value larger than the min would end up with an illegitimate solution, while decrementing all values to below min would end up with an suboptimal solution.
This way of understanding, unlike the editorial solution's explanation, does not assume the specific heuristics like the min value always get incremented and is, in my opinion, easier to relate to.
Hope this helps more or less.