The Chinese believe that after someone dies, his (or her) spirit goes to the afterlife, where it lives on pretty much like in real life. And so just like in real life, you'll need money in the afterlife - lots of it. So how does one get money in the afterlife? Through surviving relatives and friends, of course, who burn these "hell" banknotes or money to send it through (probably because Western Union has yet to open a branch in the afterlife).
The afterlife probably has one hell of an inflation problem - the denomination on these banknotes are huge! This one above is a $8 billion note (the number 8 is considered quite lucky in Chinese culture).
The Chinese don't mean to neglect their more virtuous ancestors; the "hell" in "hell money" is probably just an English mistranslation, so heavenly types get theirs too. You can read more about the Chinese afterlife at yogichen.org and (no relation to Neatorama) Sinorama magazine.