How Does a Separable Filter Work?
Steve Eddins has a nice page that explains it.

It comes down to being able to do a separate 1-D filters in each direction (requiring $MN(P + Q)$ multiplies and adds), rather than a single 2-D convolution ($MNPQ$ multiplies and adds).  Here the image is assumed to be $M \times N$ and the convolution kernel $P \times Q$.

Of course, this assumes you're doing everything in the time domain, and not using FFTs.