How do multirate systems introduce 'efficiency' in computation?
I will try to explain from my understanding.

For interpolation, the whole process can be viewed as two parts: up-samping with padding zeros, and the low-pass filtering. The zero padding will increase the length and thus the computational costs. For the low-pass filtering, there are lots of zeros, thus these multiplications could be omitted. 

For decimation, the first part is anti-aliasing low-pass filtering, and the 2nd is the down-sampling. Since only part of the low-pass filtering output is needed, it is apparent that the filtering for other points can be omitted.