The category $Mod^{E_n}_A(\mathcal{C})$ of $E_n$-modules for an $E_n$-algebra in a symmetric monoidal $\infty$-category $\mathcal{C}$ is defined in Lurie's Higher Algebra as a special case of a more general definition (just replace the little $n$-cubes operad with your favorite $\infty$-operad). The definition is rather obscure, however, and requires a lot of terminology I'm not entirely comfortable with. I was wondering if there is a more intuitive way to define/think about this category. For example, what's the best way to think of $E_n$-modules for an $E_n$-algebra in the category of spaces?

There is also a proof in Lurie that $Mod^{E_1}_A(\mathcal{C})$ for an $E_1$-algebra $A$ is equivalent to the category of bimodules for $A$ in $\mathcal{C}$, but again the proof is quite long and difficult. Is there an intuitive way to think about this result? I'd also be happy if there was a short proof for $A$ an ordinary associative algebra.