The input-output operator is just a fancy name for the function that maps the input to the output. The use of $f$, which stands for frequency, is fairly standard in this area. One very important reason to work frequency domain is important is that it is the analogue of diagonalizing an operator. The convolution is fairly messy to compute in time but in the frequency domain, it is just a multiplication operator. Also, in many applications, it is the frequency domain that is important. For example, an ideal lowpass filter would be specified by $H(f)=1$ for $|f|\leq f_0$ and $0$ elsewhere.