As a starting point, consider this brief explanation by John Baez about actions of the little disks operad on loop spaces:
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.