I can explain the pictures I usually draw to think of $A_\infty$ functors,
but I don't know if they're standard. Anyway, I'll describe what is
just a rubric for ingesting the long formulas, nothing more.

Let's consider first the Yoneda embedding $Y$, which re-thinks an object $L$
in an $A_\infty$-category $A$ as an $A$-module, or functor from $A^{op}$ to
chain complexes. So $Y_L(M) = hom_A(M,L).$

I confess that when I confront these formulas/concepts, I always think
in terms of the Fukaya category, which is very amenable to pictures
and for which the $A_\infty$ structures are geometric.

So I draw a curve on a piece of paper and label it $L$. (The curve
is literally a Lagrangian submanifold of my ${\mathbb R}^2$ piece of
paper.) When I want
to think of $L$ in terms of its Yoneda image, I draw the SAME curve, but as a
squiggly line.

So what is the data that the squiggly line gives us? For each object $M$
(a regular curve on my paper), we have the intersection points, which form a
graded vector space $hom_A^*(M,L).$ This vector space has the
structure of a chain complex (Floer), with differential given by
football-shaped bi-gons with one regular side and one squiggly side.
For a pair of other objects, $M_1, M_2,$ we get a map
$$\mu^2: hom_A(M_2,L)\otimes hom_A(M_1,M_2) \rightarrow hom_A(M_1,L),$$
and so on for all the structure of a module (section 1j, p. 19).

For the Fukaya category,
the equations 1.19 follow (for non-squiggly lines) from studying degenerations
of 1-parameter families of holomorphic polygons. Now squigglifying those same pictures
gives 1.19 for an arbitrary module, and the equations are similar for
not just modules but arbitrary functor between two $A_\infty$-categories.

What data do we have if we have *two* squiggly lines $L_1$ and $L_2$?
They should intersect at a morphism between functors (and it should have a degree).
This morphism of functores gives more data, using the Fukaya perspective.
If we added one normal line $M$, we'd have the spaces $Y_{L_1}(M)$ and $Y_{_2}(M)$,
and have a triangle which is a map between them. Higher polygons and the
relations between them (by considering one-parameter families) should
give you all the equations and give you a hint as to verify them.
(But no promises!)

Hope that lengthy and pretty vague description was worth our time.
(Oh, geez, this was a March 11 question? Probably stale by now!)