The result in question is a corollary to proposition 5.4.5.15 in Higher Algebra

> **Proposition 5.4.5.15.** Let $M$ be a manifold and let $C^⊗$ be an $∞$-operad. Then composition with map $\mathrm{Disk}(M)^⊗→\mathbb{E}^⊗_M$ of Remark 5.4.5.8 induces a fully faithful embedding
$$θ : \mathrm{Alg}^{nu}_{\mathbb{E}_M}(C) → \mathrm{Alg}^{nu}_{\mathrm{Disk}(M)} (C)\,.$$
The essential image of $θ$ is the full subcategory of $\mathrm{Alg}^{nu}_{
\mathrm{Disk}(M)}(C)$ spanned by the locally constant
objects.

When $M=\mathbb{R}^n$, the $\infty$-operad $\mathbb{E}_M$  is just the $\mathbb{E}_n$-operad (the small $nu$ on top is just saying that we're talking about  "non-unital" algebras).

Be careful that the book does not use the language of factorization algebras, but if you compare the definitions you'll see that there is a conservative functor from locally constant factorization algebras on $M$ to locally constant $\mathrm{Disk}(M)$-algebras, that is a left inverse to the functor of proposition 5.4.5.15.