A lot of consoles do not offer composite sync, only composite video. C-video carries the C-sync information, along with a lot of extra crap many monitors don't need, and the result is an unstable display or a completely blank display. The LM1881 is a cheap ($3) chip that takes a C-vid signal and strips out the unecessary video data, resulting in a clean composite sync signal suitable for most monitors.
There's not much to it. Simply connect the chip as shown.