Physical meaning of Transpose of an Operator in Quantum Mechanics?

From a physical point of view, the only relevant operators are those that are "symmetric" (in fact, self-adjoint). These operators have the properties of remaining unchanged under a "transposition". Any other operator that does not satisfy this property is not observable.

Having said this, there are some interesting cases, like the one mentioned in the OP, like unitary operators. These can be given the physical meaning of transformations, in a rather broad sense. More generally, one can consider more general objects, like partial isometries. A partial isometry $V$ maps a subspace of a Hilbert space into another subspace of the same dimension. Hence, every unitary is a partial isometry, but not every partial isometry is a unitary. If $V$ maps $K$ to $K'$, then it is easy to see that $V^T$ maps $K'$ back to $K$. For any bounded operator $A$ on a Hilbert space, one can find a positive operator $|A|$ and a partial isometry $V$ such that

$$A = V|A|$$

This is known as the polar decomposition of $A$. Using this, one then finds

$$A^T = |A|V^T$$

which can help to get an idea of what the transposition is doing on the operator $A$.