How did people come up with the formula $a^3+b^3=(a+b)(a^2-ab+b^2)$? Every resource that I've read proves the formula
$$ a^3 + b^3 = (a+b)(a^2-ab+b^2) \tag1$$
by just multiplying $(a+b)$ and $(a^2 - ab + b^2)$.
But how did people come up with that formula? Did they think like, "Oh, let's just multiply these polynomials, I don't know why, let's just do it." I don't think that people just pointed a finger at the sky and came up with that formula.
So, how to prove $(1)$?

Alternative approach:
Stealing the insight from the comment of Ethan Bolker.
Mathematicians discovered that
$$(1 + x + x^2 + \cdots + x^n) \times (1 - x) = 1 - x^{n+1}. \tag1 $$
For $0 \neq a,b$, 
$a^3 + b^3$ can be rewritten as
$$a^3 \times \left[1 - \left(\frac{-b}{a}\right)^3\right]. \tag2 $$
Setting $~\displaystyle x = \left(\frac{-b}{a}\right)~$ (1) and (2) collectively imply that
$$a^3 + b^3 = a^3 \times \left[1 - \frac{-b}{a}\right] \times \left[ ~1 + \frac{-b}{a} + \left(\frac{-b}{a}\right)^2 ~\right]$$
$$ = a^3 \times \left[\frac{a + b}{a}\right] \times \left[\frac{a^2 - ab + b^2}{a^2}\right] = (a+b) \times (a^2 - ab + b^2).$$
