Adding my two cents to Mike's answer. Here is a way to show that $f$ establishes a Morita equivalence as described in the question iff it gives rise to an equivalence between the Cauchy completions. It depends on few facts some of which are not trivial.

Let $\mathbf{Prof}$ denote the bicategory whose objects are categories $A, B \cdots$ and whose morphisms are profunctors $F : A \nrightarrow B$.

A presheaf category $\hat{A}$ is $\mathbf{Prof}(A, I)$, where $I$ is the terminal category.

A functor $f : C \to \hat{A}$ can be regarded as a profunctor $F : A \nrightarrow C$, and $\mathrm{Lan}_yf$ is the functor $-\circ F : \mathbf{Prof}(C, I) \to \mathbf{Prof}(A, I)$.

Every cocontinuous functor between presheaf categories comes from a profunctor. It follows that $(-\circ F)$ is an equivalence of categories if $F$ is an equivalence in $\mathbf{Prof}$.

In $\mathbf{Prof}$ there are equivalences $A \simeq \bar{A}$ between a category and its Cauchy completion. Composing $F$ from both sides with these equivalences we get a profunctor $\bar{F} : \bar{A} \nrightarrow \bar{C}$. Immediately, $F$ is an equivalence if and only if $\bar{F}$ is an equivalence.

Every morphism in $\mathbf{Prof}$ whose target is a Cauchy complete category and which has a right adjoint converges to a functor. Hence, if $\bar{F}$ is an equivalence, then it comes from a functor $\bar{A} \to \bar{C}$ which is equivalence in $\mathbf{Cat}$.

Adding my two cents to Mike's answer. Here is a way to show that $f$ establishes a Morita equivalence as described in the question iff it gives rise to an equivalence between the Cauchy completions. It depends on few facts some of which are not trivial.

Let $\mathbf{Prof}$ denote the bicategory whose objects are categories $A, B \cdots$ and whose morphisms are profunctors $F : A \nrightarrow B$.

A presheaf category $\hat{A}$ is $\mathbf{Prof}(A, I)$, where $I$ is the terminal category.

A functor $f : C \to \hat{A}$ can be regarded as a profunctor $F : A \nrightarrow C$, and $\mathrm{Lan}_yf$ is the functor $-\circ F : \mathbf{Prof}(C, I) \to \mathbf{Prof}(A, I)$.

Every cocontinuous functor between presheaf categories comes from a profunctor. It follows that $(-\circ F)$ is an equivalence of categories iff $F$ is an equivalence in $\mathbf{Prof}$.

In $\mathbf{Prof}$ there are equivalences $A \simeq \bar{A}$ between a category and its Cauchy completion. Composing $F$ from both sides with these equivalences we get a profunctor $\bar{F} : \bar{A} \nrightarrow \bar{C}$. Immediately, $F$ is an equivalence if and only if $\bar{F}$ is an equivalence.

Every morphism in $\mathbf{Prof}$ whose target is a Cauchy complete category and which has a right adjoint converges to a functor. Hence, if $\bar{F}$ is an equivalence, then it comes from a functor $\bar{A} \to \bar{C}$ which is equivalence in $\mathbf{Cat}$.