The conversation is conducted in American English slang, and is about a $25 bet - which is refused because "you're into me for $475" - ie., you owe me $475. If the bet were taken, that would add up to a total debt of $500 - half-a-string, so "a string" is $1000.
This does not seem to be a term in wide use. Perhaps the author coined it, or maybe its popularity has been limited in time, location or social class.
There are many, many slang terms for various sums of money in British and American English. A more common term for $1000 (or £1000) would be "a grand". A pony (£25) or a monkey (£500) are rarely used these days, except perhaps by writers trying to inject a little atmosphere into their pages.