The greatest common divisor between $2$ integers is the largest (in absolute value) integer that divides both integers.
If we consider two integers $a$ and $b$ and if we assume that $d$ is their greatest common divisor then there exists integers $m$ and $n$ such that $a = md$ and $b = nd$. This then implies that $d$ divides the sum $a+b$ since $a+b = md+nd = (m+n)d.$
In your case, the sum of $-90$ and $100$ is $10$. So, the greatest common divisor between $-90$ and $100$ must also divide $10$. The greatest divisor of $10$ is itself (as is the case with any integer up to absolute value) and so it is natural to check whether or not $10$ divides both $-90$ and $100$. If it divides both then we are done since $10$ is also the largest integer that divides $100+(-90)=10.$
Indeed, $10$ divides both $-90$ and $100$ and so it must be their greatest common divisor.