When Dr. Shewhart was first developing the techniques of SPC, he needed to find a straightforward, yet reliable way to find out whether or not a process showed any signs of uncontrolled variation, also called special cause variation.
If a process exhibits any special cause variation then it is not in a state of statistical control. As such, it is not possible to make any predictions about how the process will perform in the future, based on how it has performed in the past. It is also impossible, therefore, to calculate the capability of the process in any meaningful way, or to attempt to improve the process.
What Dr. Shewhart needed was a simple test to indicate the presence of a special cause, which would be reliable enough to catch almost all special causes, but not so sensitive that it produced a lot of false alarms. Based partly on his statistical knowledge, and partly on his own empirical observations, Dr. Shewhart established that ±3 sigma would be an appropriate level at which to place control limits.