The first use of weeaboo can be traced back to this comic. (The comic is called Perry Bible Fellowship, its fantastic. Read it.) Sometime very early in the lifespan of 4chan, Wapanese (I have no idea what this word means but I think its like wigger but for whites that act like Jap stereotypes) gained popularity and sometime in late 2005 it became wordfiltered to weeaboo, both as a nod to the effect it had on threads similar to how it was received in the comic and to try and reduce that effect. By this time Wapanese was pretty much a dead word, but weeaboo took on something of a meaning. It came to refer to Japanophiles who liked Japan more than their mother nation to the point of spitting on their culture despite knowing nothing of Japan outside of entry level anime. It was typically used against people from sites like gaiaonline and just whomever /a/ didn't like. You need to remember though, at the time some things that are thought of as something only weeaboos do like go to cons or watched dubbed anime were things done by /a/. 4chan had even had its own panel at Otacon until about 2007 (when they were banned) and sub groups were not nearly as prolific as they are now.
This meaning still lives on, but only in communities like 4chan's /a/ or /jp/; as I mentioned, its meaning has changed and the people it refers to have changed. That being said, weeaboo also has another far more widespread meaning, thanks to the ironic style of writing common on 4chan and its eternal summer, normals co-opted it to refer to any kind of Japanophile, and eventually any person who enjoys media that is off from the American mainstream and uses a noticeably Japanese style unironically.
Works that have a Japanese style but are very western like The Big O or Cowboy Bebop often avoid this stigma, as does pretty much anything shown in America in the 1990s to the early 2000s. For one reason or another, this does not include Naruto, Bleach, One Piece, InuYasha, and Sailor Moon. In avoiding this stigma, The Big O and Cowboy Bebop have managed to draw the ire of Western otaku communities simply by being recommended by the same groups that ostracize them for liking other anime, even if its something like Black Lagoon.
Full disclosure: I used to go on 4chan often before alot of things happened in my life, one of them being coming here. Also, I love Cowboy Bebop and it really makes me sad whenever someone acts like I killed their dog and proceed to strip me of all my otaku cred for ever suggesting that Knocking on Heavens Door had good animation.
Sorry for the wall of text. If I got anything wrong please say something, I'm not saying what I wrote it the absolute truth, just my observations.