In programmers terms, support code is the underlying program code that a system runs on; it's 'foundation'. The rest of the sysytem relies on this program code to actually work -- support code takes 90% of the code time in any given project.
For example: I have a whizbang idea for a new auction site; I'll call if 'fBay' or something silly like that. The core of the system, the bidding system, is easy to write. It's the other stuff, like user managment, style sheets and checking for errors that Can't Happen(TM) that takes up all the time and effort.
It's also called 'support code' because it's what you end up supporting for years and decades after the original project has been abandoned and left inthe dust.