I understand that the function $\Theta(z|\tau)$, with its definition $$\Theta(z | \tau) := \displaystyle\sum_{n = -\infty}^{\infty} e^{\pi in^2\tau} e^{2\pi inz},$$ satisfies the product formula $$\Theta(z|\tau) = \prod(z | \tau) := \prod_{n = 1}^\infty(1-q^{2n})(1 + q^{2n-1}e^{2\pi iz})(1 + q^{2n-1}e^{-2\pi iz}).$$
A proof of this from Stein and Shakarchi's Complex Analysis uses the fact that both function satisfy same two quasi-periodic conditions in $1$ and $\tau \in \mathbb H$ and have common (simple) zeroes. I understand the entire proof given in the book. I also found another proof from this website which I only briefly skimmed through.
However, I am unsure how people may have come up with the product formula in the first place. Where did they get the intuition, and how did they develop the actual theorem (that $\Theta = \prod$) from their intuition? I would also be grateful if you could provide some historical context, if you have any knowledge of that.
I would greatly appreciate any input, however small.