I'm pretty sure someone came up with this answer but never bothered to write an answer. I bother.

Recall that if $\mathcal{D}\subset \mathcal{C}$ is a reflective subcategory, then the essential image of the inclusion $i\colon \mathcal{D}\hookrightarrow\mathcal{C}$ consists of those objects $X$ such that for any map $f\in\operatorname{mor}\mathcal{C}$, the map $\operatorname{Map}_{\mathcal{C}}(f,X)$ is an isomorphism (in the homotopy category of Kan complexes). (For a proof, see, e.g., this MO question. The answer treats the ordinary categorical case but the proof works perfectly fine in the $\infty$-categorical case.) Since the Yoneda embedding preserves limits, it follows that the essential image of $i$ is closed under limits in $\mathcal{C}$. We may now conclude by observing that the reflector restricts to an equivalence between the essential image of $i$ and $\mathcal{D}$.

Recall that if $\mathcal{D}\subset \mathcal{C}$ is a reflective subcategory, then the essential image of the inclusion $i\colon \mathcal{D}\hookrightarrow\mathcal{C}$ consists of those objects $X$ such that for any map $f\in\operatorname{mor}\mathcal{C}$, the map $\operatorname{Map}_{\mathcal{C}}(f,X)$ is an isomorphism (in the homotopy category of Kan complexes). (For a proof, see, e.g., this MO question. The answer treats the ordinary categorical case but the proof works perfectly fine in the $\infty$-categorical case.) Since the Yoneda embedding preserves limits, it follows that the essential image of $i$ is closed under limits in $\mathcal{C}$. We may now conclude by observing that the reflector restricts to an equivalence between the essential image of $i$ and $\mathcal{D}$.