Because the set of multiples of $\gcd(a,b)$ is exactly the set of values $ax + by$ for all integers $x, y$

Because adding $\frac{b}{\gcd(a,b)}$ to $x$ and subtracting $\frac{a}{\gcd(a,b)}$ from $y$ preserves the equation

Both (a) and (b) explain it

It's a formula that works empirically but has no deep reason