Originating on Japanese imageboard 2channel, the '>' sign was used for quotations. When '>' preceeded a line, it was coloured in light green. Later carried over to 4chan and by extension all other futaba-imageboards, it was still used for highlighting text or quoting other users. Also notoriously used to show that you were an experienced user (>newfags can't green text).
Lately (late '09-early '10), a phenomenon originating from 4chan board /v/ (video games) spread - implying. It is used to sarcastically respond to another user who posted a false or biased statement. Example:
Poster #1: Pinkie Pie is the best pony!
Poster #2:
>implying she is
Another fenomenon called "greentext stories" originated on the 4chan /r9k/ board, where people would post stories in the format you just posted. Most of the time, these stories would have a common theme (something like "AWKWARD GREENTEXT STORIES, GO!" in the OP).
http://i.imgur.com/GORru.png