Deligne's conjecture (the little discs operad one) Deligne's conjecture states that the Hochschild cochain complex of an A-infinity algebra is an algebra over the operad of chains on the topological little discs operad.
Of course the conjecture has been proven, but proofs aside -- what are some moral reasons why we should believe in Deligne's conjecture?
 A: Section 2.5 (third paragraph) in Lurie's DAG 6 (on his web page) has a general explanation.  For a space to have an n-fold loop space structure is equivalent to having an action of the E[n] operad, and taking E[n] Hochschild cochains amounts to looping.  For the case under consideration, E[1] is equivalent to A-infinity, and the little discs operad is a model of E[2].
A: You should read the (extremely drafty) draft of "Blob homology" by me and Kevin Walker, which includes a nice topological explanation of Deligne's conjecture, and a higher-dimensional generalization.
Unfortunately we don't say "Deligne's conjecture" in exactly the way you did, but the essential thing is that we get a chain map $$C_*(Diff_M)(x)B_*(M;C)\to B*(M;C)$$ for any $n$-manifold $M$ and (roughly speaking) $n$-category $C$. Restricting to $M=S^1$ and $C$ an $A$-infinity algebra gives you what you want, after a few more steps of translation.
A: As a starting point, consider this brief explanation by John Baez about actions of the little disks operad on loop spaces:
http://math.ucr.edu/home/baez/week220.html
Now consider the figure on page 35 of these slides -- http://canyon23.net/math/talks/GTC%20200905b.pdf
Up to homotopy, the little disks space is equivalent to the "big bigons" space.  The outer bigon is almost entirely filled by the inner bigons (so they are as big as they can be).  We can think of this as describing a sequence of operations (one for each inner bigon) which transforms the lower half of the outer bigon into the upper half of the outer bigon.  i.e. cut out the lower boundary of a lowest inner bigon and replace it with the upper half of that inner bigon, and so on for each inner bigon.
We think of Hochschild cochains as a derived Hom from the regular bimodule to itself.  If each inner bigon is labeled by a Hochschild cochain, then composing these elements of various Hom spaces, in a manner tracking the topological operations in the previous paragraph, gives a Hochschild cochain associated to the outer bigon.
So far we have described an action of 0-chains (single points) of the big bigons operad to Hochschild cochains.  If we two points in the operad connected by an arc, then the maps associated to the endpoints are not equal, but they are chain homotopic via a homotopy determined by the arc.  And so on for k-chains in the operad.
I'm not sure how hard it would be to turn the above ideas into a proof for the usual Hochschild cochain complex, but for the homotopy equivalent blob complex one can give a proof along these lines.  This proof (for the blob complex) generalizes to higher dimensions, where we replace the boundary of a bigon (two intervals) with any two n-manifolds glued along their boundary.  Actions of the little n-cubes operad come from the special case where all the n-manifolds are n-balls.
A: Here is a super simple way to think about it: The Hochshcild cochains on $A$ is derived $Hom$ over $A \otimes A^{op}$ of $A$ into$ A$. This has two multiplications, given by composition and multiplication in the target respectively.
These commute, or intertwine, so the result has an action of the tensor product of two $A$-infinity operads, which is (at least if you chose the right $A$-infinity operads) an $E_2$ operad. 
