I'm just trying to understand the following definition:

Definition 3.3.3.8 in Higher Algebra by J. Lurie defines the $\infty$-operad of $O$-module objects, and says the following:

Let $O^\otimes$ be a unital $\infty$-operad and $C^\otimes \to O^\otimes$ a fibration of generalized $\infty$-operads. We let $Mod^O(C)^\otimes$ denote the fiber product $$\overline{Mod}^O(C)^\otimes \times_{^p Alg_{/O}(C)} (O^\otimes \times Alg_{/O}(C))$$

My question is what does an object in here look like?

Here is my rough attempt at understanding what is going on: First an object of $\overline{Mod}^{O}(C)^\otimes$ consists of the data $(v, F)$ of an object $v \in O^\otimes$ and a functor $F:$ $ _{v}K_O \to C^\otimes$ (where $_{v}K_O$ denote those semi-inert maps in $O^\otimes$ that start at $v$) sending a semi-inert map $f:v \to y$ in $O^\otimes$ to an object $F(f$) of $C^\otimes_y$, such that if $y \to z$ is inert, then $F(f) \to F(v \to z)$ is inert in $C^\otimes$.

So my guess is that an object of $Mod^O(C)^\otimes$ is a pair $(v, A)$ where $v \in O^\otimes$ and $A \in Alg_{/O}(C)$. Or perhaps it is something like $(v, F, A, \theta)$ where $\theta$ is an equivalence between $(v, F)$ and $(v, A)$ in $^p Alg_{/O}(C)$. Or perhaps it's something else entirely.

What is, intuitively/philosophically, the role of the semi-inert morphisms, i.e. why do we really have to use them to make this definition?

I think I understand how an object in here can be viewed as generalization of the classical notion of a module over an algebra. In chapter 4 when Lurie defines the $\infty$-operad $\mathcal{LM}^\otimes$ and defines an $\infty$-category $LMod(C)$ (for $C$ a monoidal $\infty$-category) of left modules over the associative operad as $Alg_{\mathcal{LM}/\mathcal{Ass}}(C)$, and then example 4.2.1.18 shows how to think of an object in this category as a generalization of the notion of a module. I assume a similar analysis holds for objects in $Mod^O(C)$ e.g given $(v, A)$, say $v\simeq (v_1, ..., v_n)$, where the $v_i \in O$ then $A(v_1), ..., A(v_n)$ have various multiplications maps between them, controlled by the operad $O$. If this more explicitly spelled out in Higher Algebra let me know.