Given $T:V\to V$ linear and $V$ being an inner product space, we define $T^*$ by a linear operator on $V$ such that $\langle Tx,y\rangle=\langle x,T^*y\rangle$ for each $x,y\in V$.
We later see that, for finite-dimensional inner product spaces, adjoint exists and, in fact, whenever an adjoint exists is unique.
For finite-dimensional space we can draw a correspondence between conjugate transpose of a matrix and adjoint of a linear operator, taking into consideration an orthonormal basis of $V$ and representing $T$ with respect to that basis.
Now I think the motivation behind such a definition is finding a linear transformation version of the conjugate transpose. We know that if $T$ is a linear operator on inner product space $V$, and $\beta$ be an orthonormal basis of $V$, then the corresponding matrix $A_{ij}=\langle T\alpha_j,\alpha_i\rangle$.
So, naturally if $B$ is the conjugate transpose of $A$, then $B_{ij}=\overline{ A}_{ji}=\overline{\langle T\alpha_i,\alpha_j\rangle}$. So, naturally, a question arises if there exists $U$ linear on $V$ such that $B$ is the matrix of $U$ with respect to the given orthonormal basis. Then $B_{ij}=\langle T^*\alpha_j,\alpha_i\rangle=\overline{\langle T\alpha_i,\alpha_j\rangle}=\overline {A}_{ji}$.
This, I think, motivated the definition of adjoint.
Best Answer
A slightly different, but essentially equivalent, way of thinking of it is that the adjoint is the linear operator on the dual space $V^*$ induced by $T$.
Consider the dual space $V^*$, consisting of linear maps $V \to \Bbb{R}$. Then there is a map $T^*$ on $V^*$ defined as follows: for $\lambda \in V^*$, $T^*(\lambda)$ is the linear functional given by: $$ \Big( T^*(\lambda) \Big)(v) = \lambda \Big( T(v) \Big) $$ If you choose a basis for $V$, for which $A$ is the matrix of $T$ in that basis, then the adjoint matrix (the transpose) is the matrix of $T^*$ in the dual basis.
The connection with your explanation is this: if you have a nondegenerate inner product on $V$ and $V$ is finite-dimensional, then there is an isomorphism between $V$ and $V^*$ using the inner product. For a vector $v$, define a linear functional $\lambda_v$ by the formula $\lambda_v(w) = \left<v,w\right>$. If you use this to identify $V$ and $V^*$, (and if the basis is orthonormal), then the basis and the dual basis are the same, and you can think of $T^*$ as an operator on $V$.