AMO tracks clicks by using a tracking pixel. For Search Engine ads like Google or Bing, the URL of the ad is wrapped by a tracking URL (pixel.everestech.net....more parameters). When the searcher clicks on the ad, the user is first directed to Adobe servers where we gather all the tracking information and then the user is sent to the advertisers site.
I can definitely supply more detailed and technical information, and speak as deeply as you like on this. Just let me know what questions you have.