As written in the paper that you cite, this construction originates from two papers of McClure–Smith, and a geometric interpretation of the table reduction morphism is given in:

Clemens Berger and Benoit Fresse. "Une décomposition prismatique de l'opérade de Barratt-Eccles." (French) *C. R. Math. Acad. Sci. Paris* 335 (2002), no. 4, pp. 365–370.

In particular, I think Lemme 3.3 is enlightening. Let me very quickly summarize it.

Recall that $\mathcal{E}(n) = \Bbbk[E\Sigma_n]$ is the Barratt–Eccles operad, where $\mathcal{E}(n)_d = \Bbbk[\Sigma_n^{d+1}]$. It is the linearization of a simplicial operad $\mathcal{W}$. On the other hand, $\mathcal{X}$ is the surjection operad, where $\mathcal{X}(n)_d$ has for basis the set of surjections $u : \{1,\dots,n+d\} \to \{1,\dots,n\}$ such that $u(i) \neq u(i+1)$ for all $i$.

For a surjection $u \in \mathcal{X}(n)_d$, let $d_k = \# u^{-1}(k)$. There is an associated prism $\tau_u : \Delta^{d_1-1} \times \dots \times \Delta^{d_n-1} \to \mathcal{W}(n)$ defined combinatorially. Among all these prisms, there is a maximal one, called the fundamental simplex. Then:

Lemma 3.3. The map $TR : \mathcal{E} \to \mathcal{X}$ is characterized by the following properties:

- it is a chain map;
- if $\sigma$ is the maximal simplex of a prism $\tau_u$, then:
$$TR(\sigma) = \begin{cases} u & \text{if } \sigma \text{ is the fundamental simplex of } \tau_u; \\ 0 & \text{otherwise.} \end{cases}$$