It’s basic number theory.
We start with the positive integers - the counting numbers, and the operation of addition.
When we “add” two integers we necessarily get a third integer. The set of integers is closed under addition.
we create an ordering on the integers, a relationship so that for each integer we can say which come earlier (are “smaller”) and which come later (are “bigger”).
We note that zero is a special value under addition.It maps each integer back to itself. It’s called the “identity” under addition.
We define an operation inverse to addition and call it subtraction. We note that both addition and subtraction share the same identity element: zero.
Unfortunately our set of positive integers isn’t closed under subtraction. 3 subtract 5 isn’t in our set.
So we extend our set: we invent a new concept of “negative numbers” and include them in our set so now our set of (all) integers is closed under both addition and subtraction.
3 subtract 5 is one of those negative integers. Let’s call it Simon. Simon has the special property that Simon add 2 is the zero. In recognition of this fact we label Simon as “-2” and speak it as “minus two”.
in fact we can pair each positive integer in our original set with a friend in the new set of negative numbers, so that when we add any positive integer to its friend the answer is the additive identity, zero.
This gives a convenient labeling for the negative numbers, each as the inverse under addition from the identity element (zero) of a different positive numbers. Writing them pairwise we have 1 and -1, 2 and -2, 3 and -3 etc.
Now we move on to a new operation called multiplication. What properties does it have that will make it useful? Our set of integers is closed under multiplication, but not under the inverse operation “division” - we’d need to invent the idea of rational numbers (fractions) for that - but we can do that later, not now. To be useful we require that multiplying any number in our set by the additive identity (zero) gives zero.
when we look at the structure of our set under multiplication, to be self-consistent (so as not to be able to prove logically things like 1 = 0, which would make our system useless), it turns out that -1 x -1 has to give the same answer as 1 x 1.
similarly Simon x Simon has to give the same answer as 2 x 2.
So the answer is that it’s implicit in the structure of the number system we invented. If minus x minus wasn’t plus we couldn’t use arithmetic for anything useful as it could “prove” nonsense answers like 1 = 0.