So basically what he did in mathematical terms is

1. He differentiated the function, found the change in array keys like +100, -100 etc. :-

   arr[lower-1]+=sum;
   
   if(upper < n) arr[upper]-=sum; 
   

2. Then he integrated the derivated function to get the original function value at different points and found the max value:-

        for(int i=0;i<n;i++){
                            temp += arr[i];
                               if(temp> max) max=temp;
                   }