While I personally detest all that AIM (namely AOL) stands for, I'm forced to use it by a simple lack of options. The majority of my companions are the non-technical type, and prefer the user-friendly environment that I abhor, rather than the slight learning curve associated with ICQ (which offers away messages, too) or other IM applications. So, for the sake of my social life and chances of meeting a potential significant other, I use AIM.
While the most common use for away messages is simply to inform would-be IMers that said away person is AFK for the time being, and may include the reason for absence. However, being a member of the youth of today (circa 2001) I both have used and have had away message innuendo used on me. Often associated with talking to a member of the opposite sex, or whatever your sexual preference may be, this delicate art is a form of subtle suggestion by means of away messages. If aforementioned person is actually away, varies from case to case, but the general purpose of this tactic is to imply something or move the conversation to a certain direction, usually one involving the two parties involved engaging in somewhat a romantic and/or sexual activity.
A form of a shy individual being able to get a point across while making it seem as if it was just a harmless away message, intended for all to see and read, while its true purpose is one of wooing the one it's directed at. Simple poems often work well, a slight mention of affection, a past lover, deep emotions, or anything of the like often prompt the one reading it to inquire what context the away message was in. Once that happens, the plan's a success. From then on, it's a series of normal coquettish conversation until the idea of the two involved making the endeavor suggested by the one with said away message.
While some may see this as a devious way to trick someone into becoming interested in you, I've found that it's simply a way for those too reserved to speak their mind to make their point heard and understood. The Anonymous Coward of love.
Note: This can also be applied to AOL Instant Messenger profiles.