Very short..
Because it is easier to measure waves (voltage, power...) instead of total voltages as in low frequency circuits with oscilloscope and voltmeters.
I think there are several other post dealing with this subject, where it is explained in more detail. Just search here on the site.
One of the main aims in microwave is providing a matching of various circuits. And with S-parameters we can easy measure a level of mismatch, leflected wave, gain...
There are some different systems do describe electrical networks. For example H-parameters, Z-parameters and so on. But S-parameters are simply voltage ratios, therefore they can be relatively easily measured with high accuracy. The well known Network Analyzers are devices that do this job for us. This is why S-parameters are so popular in RF and microwave engineering. They were introduced by Kurokawa around 1965 and a couple of decades later started to be used in practical engineering mainly due to HP (now Agilent) developments of the theory and practice of S-parameters usage. Every RF and microwave engineer and technician must know how to use S-parameters.