...if adding a negative number yields a smaller number, then subtracting a negative number would do the opposite by increasing that number.... But . . . that's not an explanation.
It might not be an explanation you accept, but saying that "doing the opposite ought to lead to an opposite result" is
an explanation, and a fairly reasonable one.
The explanation provided here earlier was more logical than mathematical: the result must be "this" or "that", and "that" leads to a logical contradiction, so the result then must be "this". (A mathematical proof would involve graduate-level field theory.) Your father has provided you with another logical explanation; namely, if "this" leads to "the other", then "not this" ought sensibly to lead to "not the other". And there are various other explanations and mental pictures, such as those provided here
, and here
If none of these "works" for you, then you may just need to accept the rule for now, and give yourself some time to become comfortable with its practice.