To give a more precise answer, I think the easiest way to define the adjoint is to first find out what it has to be, and then check that the definition we have found indeed works.
So suppose there is a left adjoint $L_c$ to the evaluation functor. Given that $L_c$ is a left adjoint, it has to preserve colimits, and both the category of sets and the presheaf category happen to have colimits. But we also know that any set $X$ can be obtained as a colimit $X\simeq \coprod_{x\in X}1$, where $1$ is the singleton. So we necessarily have that $L_c(X) \simeq \coprod_{x\in X}L_c(1)$ in the presheaf category $\widehat{C}$. So it suffices to determine $L_c(1)$ to determine the entire functor.
The adjunction $\operatorname{Hom}_{\widehat{C}}(L_c(X),D) \simeq \operatorname{Hom}_{\mathbf{Set}}(X,D(c))$ specialised to $X = 1$ gives then $\operatorname{Hom}_{\widehat{C}}(L_c(1),D)\simeq \operatorname{Hom}_{\mathbf{Set}}(1,D(c)) \simeq D(c)$. This lets us guess what $L_c(1)$ should be as we happen to know an presheaf satisfying exactly this equation for any other presheaf $D$. This is given by the Yoneda lemma :
If we denote $y: C \to \widehat{C}$ the Yoneda embedding defined by $y(c) = \operatorname{Hom}_C(-,c)$, then the Yoneda lemma states that $\operatorname{Hom}_{\widehat{C}}(y(c),D)\simeq D(c)$, which is exactly what we wanted.
We can then reconstruct the entire $L_c$ from these observations, by taking $L_c(X)\simeq \coprod_{x\in X}y(c)$. Now we just have to check that our guess was the right one, and $L_c$ is indeed the left adjoint to the evaluation functor. This is given by the following isomorphisms:
$$
\operatorname{Hom}_{\widehat C}
\left(
\coprod_{x\in X} y(c), D
\right)
\simeq \prod_{x\in X} \operatorname{Hom}_{\widehat{C}}(y(c),D)
\simeq \prod_{x\in X} D(c)
\simeq D(c)^X
\simeq \operatorname{Hom}(X,D(C))
$$
