If I give you a dollar for a burger, then I've lost a dollar and gained a burger, and you've gained a dollar and lost a burger. Assuming this was a fair trade (that dollars and burgers are of approximately equal value), then as a result of the transaction we've simply rearranged who has which good, and no additional value was created in the process.
Of course, when I give you a dollar for a burger that's not really a zero sum transaction, because otherwise we wouldn't be motivated to enter into the transaction in the first place. I give you a dollar because I want the burger more than I want the dollar, and if you accept the trade it's because you want the dollar more than you want the burger, so in a fair exchange we both feel like we've come out ahead. In other words, there is some additional value in our fair exchange that is not accounted for in the burgers and the dollars alone. But if our bookkeeping method only counts burgers and dollars, then it's not accounting for the value that accrues in our transactions.
A very interesting idea. Not sure how I feel about this in practice, it's very cool to think about.