Why does removing the first digit larger than the next one minimize the result? Numbers are compared left to right. The leftmost digits have the most impact. A large digit early hurts more than a large digit late.
When you see a peak (digit followed by smaller digit), removing the peak always reduces the number. Counter-example: in '', removing any non-leftmost digit leaves a larger number than removing ''.
This greedy choice is safe because you fix the leftmost problem first, which has the biggest effect on the final number.