Grand has been a slang term for $1000 since at least 1915, although it may have been in use for years before our first record of it. Back in those days $1000 was worth the equivalent of $47,000 (or more by some measures), certainly a grand amount. Other than this, we have little knowledge as to how this term may have come about.
Sometime between 1915 and 1940 'grand' became shortened to 'G' or 'G-note', although it continued to be used in its original form. When American soldiers went over to England during Wold War II they brought their slang with them, and soon grand came to mean £1000 as well. Over time grand spread to most, if not all, English speaking countries.
In the 1940s grand came to be used to mean a thousand of almost anything, but over time it has come to be used almost exclusively for money. It is still in common use today in most English speaking countries, and is used in a diverse range populations, from the ghetto to WASPs.
It is interesting to note that grand has had an even longer life as a popular slang term than does cool. Few words have this distinction, although it is worth noting that thou, short for thousand, is even older, first appearing in print in 1869.