Dirac notation makes this more transparent.

-representation of an operator in position space. It would be a matrix if the states were discrete.

Let us examine the two state system.

Both V and W can be represented wrt the x-y basis

So the argument is that an operator that takes a state vector and transforms it into another state vector is defined in terms of how much of each basis element is changed and the intial state amplitudes define how much of any basis state there is. Thus the initial vector transforms into a new vector via an nxn matrix where n is the dimension of the space. QM may restrict operators. In some cases they need to be linear or have an inverse. But they can be viewed as analogous to matrices.