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!)
As far as I know, the prototypes of obstruction theories in algebraic geometry originated from the more general Kodaira-Spencer theory of deformation of complex manifolds [see Kodaira-Spencer, On deformations of complex analytic structures I-II-III, Annals of Math., 1958-1960].
The crucial question was
When an "infinitesimal" deformation of a compact complex manifold $M$ (in particular, of a complex projective variety) gives rise to a "genuine" deformation of $M$, i.e. a deformation over a disk?
The answer to this question is contained in the following result, see [Kodaira, Complex manifolds and deformations of complex structures, Theorem 5.1]:
Theorem. Suppose given a compact complex manifolds $M$, and $\theta \in H^1(M, \Theta_M)$. In order that there may exist a complex analytic family $\omega \colon \mathcal{M} \to B$ such that $\omega^{-1}(0)=M$ and $(dM_t/dt)_{t=0}=\theta$, it is necessary that $[\theta, \theta]=0$ holds.
In fact, Kodaira explicitly says that "if $[\theta, \theta] \neq 0$ there is no deformation $M_t$ with $\omega^{-1}(0)=M$ and $(dM_t/dt)_{t=0}=\theta$. In this sense, we call $[\theta, \theta] \in H^2(M, \Theta_M)$ the obstruction to the deformation of $M$".
Of course Kodaira was well aware that the condition $[\theta, \theta]=0$ is not sufficient in general, since there can be higher-order obstructions, corresponding to the need of finding higher and higher truncations of the solution of the Maurer-Cartan equation governing the deformation of the given complex structure. Only if all these obstructions vanish we can hope to find our complex analytic family $\mathcal{M}$.
In Kodaira's words, "Thus we have infinitely many obstructions to the deformations of $M$. In view of this fact we call $[\theta, \theta]$ the primary obstruction".
The obstruction theories coming later in algebraic geometry, as far as I know, were build up in order to rephrase and extend Kodaira-Spencer theory in a completely algebro-geometrical setting (for instance, making possible deformation theory in characteristic $p$), in order to deform objects different from complex structures, such as coherent sheaves, subvarieties, maps, and in order to understand the difference between the deformations in the analytic sense and those in the algebraic sense ("algebraization problem").
Best Answer
Here's a view of the symplectic side of the bridge.
The Kuranishi model (see Donaldson-Kronheimer, The geometry of four-manifolds, ch. 4) goes like this. You're interested in a (moduli) space $M$ cut out as $\psi^{-1}(0)$, for some smooth but nonlinear map of Banach spaces, $\psi \colon (E,0) \to (F,0)$ such that $\delta:=D_0\psi$ is a Fredholm operator (finite dim kernel and cokernel). That means that $\delta$ is "almost" an isomorphism, and Kuranishi's principle is that one can construct a non-linear map $\kappa \colon \ker(\delta) \to \mathrm{coker}(\delta)$, such that $\kappa(0)=0$ and $D_0 \kappa =0$, and (locally near $0$) a homeomorphism $M \to \kappa^{-1}(0)$. This gives a finite-dimensional model of $M$. "Kuranishi structures" are a formalism in which one can say that $M$ is everywhere-locally given as the zeros of maps like $\kappa$.
In the case of genus 0 GW theory, $M$ is the moduli space of (say) parametrized pseudo-holomorphic maps from $S^2$ to an almost complex manifold $X$; $\psi$ is a non-linear Cauchy-Riemann operator, and, for a pseudo-holomorphic map $u\in M$, $D_u \psi$ is a linearized C-R operator - the $(0,1)$-part of a covariant derivative acting on sections of $u^\ast TX$. Its kernel can be identified with the holomorphic sections $H^0(S^2,u^\ast TX)$ of the holomorphic structure on the vector bundle $u^\ast TX$ defined by the C-R operator. Its cokernel is isomorphic to $H^1(S^2,u^\ast TX)$. If you're lucky, you have a Zariski-smooth moduli space $M$ whose Zariski tangent space at $u$ is $\ker D_u\psi$. In this case, one could take $\kappa=0$, and then the spaces $\mathrm{coker} (D_u\psi)$ form a vector bundle $Obs \to M$, which is what symplectic geometers usually call the obstruction bundle. One can now try to divide everything by $Aut(S^2)$, and quite possibly get into orbi-mathematics.
In the integrable case, still with non-singular but excess-dimensional $M$, I could write $Obs$ as $R^1\pi_* \Phi^*T_X$, where $T_X$ is the holomorphic tangent sheaf, $\Phi$ the evaluation map $\mathbb{P}^1\times M \to X$, and $\pi$ the projection to $M$. At this point, if what I've said is accurate, you and other algebraic geometers out there are better placed than me to answer your question. Is it your $(E_{-1})^\vee$?
Of course, you didn't really want to assume $M$ smooth. A place where these deformation theories are compared in generality is Siebert's 1998 paper Algebraic and symplectic Gromov-Witten invariants coincide.