Since the thread referenced isn't highly explicit, I'll rehash it here.
When Google sends you a referral, the referrer field contains information indicating which page of SERPs that visitor clicked on to arrive at your site. This is in the form of "&start=nnn", or the lack thereof.
If there's no "start=", then the referral came from page 1 (roughly speaking, you're in the top ten for that keyword, at least via that data center, at least at that moment of the day :-)
If there's a "start=0", same deal.
If there's a "start=10", then the referral came from the Google results page whose first entry is the 11th ranked listing.
Niggling point: if I set Google to give me 100 results per page, search for your site, find it at position 99 and click on it, your referral field will see a "start=0" in it for that hit. The vast majority of people use the default Google setting of 10 results per page, so this fact is unlikely to skew the results.
This technique is in some ways superior to tools that go out and query Google to determine your exact ranking.