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].