The elliptic functions are doubly periodic indeed, but they cannot serve in the analysis of 2D-signals. First they are analytic (resp. meromorphic), which implies that they satisfy the CR-equations and so cannot be used to represent an arbitrary doubly periodic "picture", e.g., the function $(x,y)\mapsto\sin^2(2\pi x-6\pi y)$. Second they necessarily have poles.
For 2D signal analysis you can use double Fourier series instead. The basis functions are suitably normalized functions
$$e_{jk}(x,y):=\exp\bigl(2\pi i(jx+ky)\bigr)\ ,$$
and you obtain the Fourier coefficients of an arbitrary doubly periodic function $$\ f:\ {\mathbb R}^2 /{\mathbb Z}^2\to{\mathbb C}$$
by integration, as in the one-dimensional case.
First of all, there is an error in the edition of Stein and Shakarchi that you are using. In later editions, the first two derivatives of $\Theta(z|\tau)$ has been replaced by the first three derivatives of $\Theta(z|\tau)$.
Here's my attempt to answer your question. I am not sure whether it is satisfactory because the expression of $c_\tau$ also includes three constants, $e_1$, $e_2$, and $e_3$, defined as
$\wp(1/2) = e_1$, $\wp(\tau/2) = e_2$, $\wp(1/2+\tau/2) = e_3$.
For completeness, I will also briefly go through the proof of equality itself.
From Corollary 1.5, we know the LHS, hereafter denoted as $L(z)$, is an elliptic function of order 2 with periods 1 and $\tau$, and with a double pole at $z=1/2+\tau/2$. (From here on I will denote $1/2+\tau/2$ as $z_0$.)
Furthermore, multiplying $L(z)$ by $(z-z_0)^2$ and letting $z\rightarrow z_0$, we can see the coefficient of the double pole $\frac{1}{(z-z_0)^2}$ is exactly 1.
We also know that $\wp(z-z_0)$ is an elliptic function of order 2 with periods 1 and $\tau$, and with a double pole at $z=z_0$. Its coefficient of the double pole is also 1. (This can be easily seen from the definition of $\wp(z)$.)
Therefore, $L(z)-\wp(z)$ is an elliptic function that is entire. It must be a constant. This establishes the desired equality.
To get $c_\tau$, we take the derivative of both sides with respect to $z$, and square both sides. We use Theorem 1.7 from the Chapter 9.
$(L')^2 = (\wp')^2 = 4(\wp - e_1)(\wp - e_2)(\wp - e_3) = 4(L-c_\tau-e_1)(L-c_\tau-e_2)(L-c_\tau-e_3)$.
Setting $z=z_0$, all $\Theta(z|\tau)$ vanish. What is left is an equation of the first three derivatives of $\Theta(z|\tau)$ at $z=z_0$, $c_\tau$, and the three constants $e_1$, $e_2$, and $e_3$.
Best Answer
Theta functions are intimately related to elliptic integrals and elliptic functions. In particular their values at $z=0$ which also go by the name thetanulls have a direct relationship with elliptic integrals.
Here is a brief summary of such key relationships. Let's start with a number $k\in(0,1)$ which is called elliptic modulus and let $k'=\sqrt{1-k^2}$ be the complementary (to $k$) modulus. We then define complete elliptic integral of first kind $$K(k) =\int_{0}^{\pi/2}\frac{dx}{\sqrt{1-k^2\sin^2x}}\tag{1}$$ and complete elliptic integral of second kind $$E(k) =\int_{0}^{\pi/2}\sqrt{1-k^2\sin^2x}\,dx\tag{2}$$ If the values of $k, k'$ are available from context then $K(k), K(k'), E(k), E(k') $ are usually denoted by $K, K', E, E'$ and they satisfy the fundamental identity $$KE'+K'E-KK'=\frac{\pi} {2}\tag{3}$$ The theta functions play a role in inverting these functions. Thus if the values of $K, K'$ are known then the values of $k, k'$ can be obtained as functions of a parameter $q$ defined by $$q=\exp\left(-\pi\frac{K(k')} {K(k)} \right) =e^{-\pi K'/K} \tag{4}$$ which is also called nome corresponding to modulus $k$. The nome $q$ is also related to the parameter $\tau$ used in definition of theta functions in your question via $$q=e^{\pi i\tau} ,\tau=i\frac{K'}{K}\tag{5}$$ and we have the following formulas $$k=\frac{\vartheta_{2}^{2}(q)} {\vartheta_{3}^{2}(q)},k'=\frac{\vartheta_{4}^{2}(q)} {\vartheta_{3}^{2}(q)},K=\frac{\pi}{2}\vartheta_{3}^{2}(q), \vartheta_{i} (q) =\vartheta_{i} (0;\tau)\tag{6}$$ These formulas also help you evaluate thetanulls in terms of elliptic integrals and moduli.
Here is deep and important result which is key to certain closed form evaluations:
A number of mathematicians (most famously Ramanujan) found closed form expressions for modulus $k_n$ corresponding to many positive integers $n$. Next comes the surprising result:
Using these theorems you can thus evaluate the values of theta functions for $z=0,\tau=i\sqrt{n}$ where $n$ is a positive rational number.