I don't see why advertisers need an extra piece of code on their site; G could already know if someone visited their site
There's more to it than that.
The tracking codes can be a lot more specific than just the fact that a user visited your site. You can create different tracking codes for different sections of your site or even individual pages, then set up remarketing lists to show different ads to different users.
For example, you could put separate tracking code on your blue widgets page and your purple widgets page, then show the blue widgets visitors followup ads about blue widgets, and the purple widgets visitors ads about purple widgets.
Or you might prefer do it the other way around, and show the blue widgets visitors ads about your purple widgets, because you know they already know about your blue widgets.
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One very productive group to target is users who abandoned their shopping carts. Here's how:
Create separate tracking codes for your shopping cart page and your thank you page. Then set up a combination list that INcludes users who put something in their carts but EXcludes all those who actually completed a purchase. The resulting list will consist of users who have a stronger than average interest in your products but left without buying something.
Keep in mind that your remarketing ads will have to compete in the auction along with everyone else's regular ads for whatever impressions they get. It takes good ads and careful targeting to make this productive.
Keep a light hand about it all. It could backfire in more than one way if you overdo things and show too many ads to the same user. Your CTR will to go down, which would make your ads less competitive, and you also risk annoying the user.
Set your frequency caps to aim for "gentle reminders", not "stalk the user".