You can learn much about your opponents or fellow players, if you know what armor they wear, what talent points they have, what professions etc. This is of great help, if you’re doing PvP, or if you want to know more about getting a high DPS in dungeons, and you know somebody of your own class, who hits hard.
To spy on the others, go to http://www.wowarmory.com/ or http://eu.wowarmory.com/, write the player name in the form and hit Search. If there are several players with the same name (people are not that original), choose the proper server.
You can also see the name of the guild, the person belongs to, and if you click on the guild link, you can learn who’s the guild master, incase you have a problem with a player and want to report it to his guild master.