Well, sometimes it is just to get you off their back whilst they finish off their morning coffee and croissant.
Sometimes it helps, like it as not, software has bugs and some of those bugs manifest themselves as memory leaks. A memory leak is where a program will allocate memory for its own use but not completely release it before allocating more memory later on, if left unchecked a program can consume vast amounts of memory and slow the system down and/or prevent other programs from executing properly. Other times it could be a program that gone into some infinite loop and hogs the CPU thereby denying CPU time to other processes.
You could try tracking down such problems but as support time is usually paid for by either the client or the software house providing the support, it is generally preferable to resolve an issue quickly rather than string it out for long periods of (costly) time.
Other times, turning it off and on again is the only way, e.g., consider your Internet connection, if it slowed down after a while you might call your ISP's support line, the first thing they usually tell you to do is turn your router off for 10 seconds and switch it back on again. This is because your connection is going down a telephone line that is logically split into pipes, each pipe can carry a certain amount of traffic and the pipe that you're connected to may be very busy, a simple disconnect and reconnect may cure the problem by the sheer fact that reconnecting will generally connect you to a less busy pipe.
Another thing you may ask, why do support guys ask you to power something off for 10 seconds before powering it back on? I'm no electronics expert but I would imagine this has something to do with allowing the capacitors to discharge fully as they can still retain charge some time after powering off, after all, they are in there to smooth out the current in the first place so removing the power for a very short period of time may not completely power down the unit hence the delay asked for by the support technician.
