I know that we need to show there are two transformations and it should satisfy the traingle equalities but I am not able to understand how does one start to go about proving it.
For example how to prove that the underlying Functor from category of categories to the category of directed graphs has a left adjoint?
$$ \mathbf{U:} \mathtt{Cat} \rightarrow \mathtt{GRP}$$
There are several scenarios where the existence of a left/right adjoint follows from some assumptions. These are the adjoint functor theorems, but I don't think you need these for the example you mentioned.
Let's just see if we can deduce a functor $F:\textbf{GRP} \to \textbf{Cat}$ that will be the left adjoint. For reference let's define $U$ on categories and functors. $U:\textbf{Cat} \to \textbf{GRP}$ is a functor which takes a small category $C$ to a directed graph by taking objects to vertices and the morphisms to directed edges. A functor between small categories takes objects to objects and morphisms to morphisms that preserve source and target. Thus $U$ sends functors to graph homomorphisms.
A natural way to construct $F$ is to take a directed graph to a category with one object for each vertex and one morphism for each path of directed edges. This is essentially making a free category on a directed graph. It should be clear what $F$ does to graph homomorphisms and that $F$ is functorial, but you should do the checks.
To construct a unit and a counit I will just try the most obvious things. To get $\eta_G:G \to UF(G)$ see that $G$ is actually a subgraph of $UF(G)$, so let $\eta_G$ be the natural embedding of $G$ into $UF(G)$. To get $\varepsilon_C:FU(C) \to C$ just map objects to themselves, and for maps you send the path morphisms of $FU(C)$ to their composition of morphisms in $C$. You do need to check that $\varepsilon_C$ is functorial.
After you've checked that these transformations are natural you can then check the triangle identities which I leave to you.