Essentially, after the completion of the loss of an arm or other limb, there is a raw set of nerves poking out (not literally, but you know what I mean). These are what connect to the automail's circuits. As stated by Vic, all of these nerves are stimulated simultaneously, causing a severe, severe shock to the person.
This pain is shown to be more obvious in the 2003 anime, lasting for quite some time after the connection. In the 2009 anime (Brotherhood), the repercussions of the event are significantly tamer.
Unfortunately, it's hard to pin this down exactly to something you might feel in real life. One common example would be hitting your "funny bone" which is actually a group of nerves in the elbow. However, this still does not do justice, as the nerves are not exposed and there are not that many of them.
The only thing biologically similar in our world is that of an exposed nerve under a tooth. If you've ever lost a tooth, and touched or injured the exposed nerve, you'll know that this pain pretty much brings you to your knees. And that's a tooth-area worth of nerves; expand that to your entire shoulder and you've got a HUGE amount of pain.
Given that the pain is akin to something few humans can ever even experience, it is no surprise that even the characters in the show (like Armstrong, shown above) cannot comprehend the pain behind the process.