understanding the definition of $\infty$-operad of module objects

I know this question is a little old but I just came across it.

Roughly, as you say, from this data you get an object $v$ of $O^\otimes$ and an algebra $A$ of $Alg_{/O}(C)$. However, you also get an action of $A$ on some object $M$ over $v$. The role of the semi-inert morphisms is as a "marking": a semi-inert morphism $v \to y$ in $O^\otimes$ marks some portion of the object $y$ (the image of $v$) as set aside for a module part, and the part outside the image describes an algebra part. Here is how this is expressed.

In the following, for the sake of convenience I am going to pretend that objects of $O^\otimes$ are literally $n$-tuples $(X_1,\dots,X_n)$ of objects of the underlying category $O$, and similarly for $C^\otimes$, so that the inert maps are given by projecting off factors. You can say the following without that assumption but it might obscure what's going on.

Your object $v$ is then a tuple $(X_1,\dots,X_k)$ of objects of $O$. Your module $M$ is going to actually be a tuple $(M_1, \dots, M_k)$, where $M_i$ is a module living over the 1-object tuple $X_i$. So for our purposes, it's simpler to just look at the case where $v = X$ is an object of $O$.

There's a special semi-inert morphism, the identity $v \to v$, which gets sent to an object $M$ in the fiber $C_X$ over $X$. For any $w$ in $O$, there is also another special semi-inert morphism $v \to () \to w$: these together describe objects $A_Y$ in $C_Y$ for any $Y \in O$. Those are going to be the module and algebra objects. A general semi-inert morphism $v \to w$ is either isomorphic to one of the form $(X) \to (Y_1, Y_2, \dots, Y_m, X)$ coming from a sequence of maps $() \to (Y_i)$ in $O^\otimes$, or $(X) \to () \to (Y_1,\dots,Y_m)$. Your functor $F$ sends these to $(A_{Y_1},\dots,A_{Y_m},M)$ and $(A_{Y_1},\dots,A_{Y_m})$ respectively by the assumption that it preserves inert morphisms.

Now the rest of the structure says that we essentially get maps of spaces $$ Map_{O^\otimes}((Y_1,\dots,Y_m), Y) \to Map_{C^\otimes}((A_{Y_1},\dots,A_{Y_m}), A_Y) $$ making $A$ into an $O$-algebra, and maps $$ Map_{O^\otimes}((Y_1,\dots,Y_m,X), X) \to Map_{C^\otimes}((A_{Y_1},\dots,A_{Y_m},M), M) $$ that give the action of $A$ on $M$. These satisfy an associativity constraint -- compatibility with composition. (You can also mix the order of the terms up but that's encapsulated by compatibility with an action of the symmetric group.)

If instead of having $v = (X)$ you had $v = (X_1,\dots,X_k)$, then you would have an algebra $A$ and a $k$-tuple of $A$-modules.