Laplace Mechanism Proof: Why this product operator?
The definition of Laplace Mechanism:
$$
\mathcal{M}_L(x, f(\cdot), \varepsilon) = f(x) + (Y_1,\ldots,Y_k)
$$
Once each $Y_i \sim Lap(\Delta f/\varepsilon)$ they are indepedent of each other. So we can calcule the joint probability of them using the product of marginals.

As we know $p_x = \mathcal{M}_L(x,f,\varepsilon)$ and $z \in \mathbb{R}^k$, thus is possible to do:

$$
p_x(z) = \frac{\varepsilon}{2\Delta f} exp\left(-\frac{\varepsilon |f(x) - z|}{\Delta f}\right) = \prod_{i=1}^k \frac{\varepsilon}{2\Delta f} exp\left(-\frac{\varepsilon |f(x)_i - z_i|}{\Delta f}\right)
$$

I think that is the answer.