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I am new in Category Theory and I am a bit confused, I would be grateful if someone could help.

Say $\mathcal{D}$ is a full subcategory of a locally small category $\mathcal{C}$ and write $I:\mathcal{D}\to\mathcal{C}$ for the inclusion. Is the functor $$\hom_{\mathcal{C}}(I(-),A):\mathcal{D}^{\text{op}}\to\mathbf{Set},$$ for $A$ an object of $\mathcal{C}$, immediately a representable?

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In general the answer is no. To prove that your functor $\hom_\mathcal{C}(I(-),A)$ is representable you have to provide an object $d \in \mathcal D$ and a natural isomorphism $$\hom_\mathcal{D}(-,d) \cong \hom_\mathcal{C}(I(-),A)\ .$$

Such isomorphism does not necessarily exists, for instance consider the case $\mathcal D=1$ the terminal category (the category with one object, one morphism) and $\mathcal D=\mathbf{Set}$ and let $I\colon 1 \to \mathbf{Set}$ be the constant functor sending the only object of $1$ in the singleton set $\{\emptyset\}$. If we let $A=\emptyset$ there is no possibility of finding a natural isomorphism of the form $$\hom_{1}(-,\bullet) \cong \hom_{\mathbf{Set}}(I(-),\emptyset)$$ because for $\bullet$ the only object of $1$ we should have a bijection $$\hom_1(1,1) \cong \hom_{\mathbf{Set}}(\{\emptyset\},\emptyset)$$ but the $\hom_1(\bullet,\bullet)$ has one element while $\hom_{\mathbf{Set}}(\{\emptyset\},\emptyset)$ is empty, so such a bijection cannot exists.

Hope this helps.

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