Ugh. Don't get me started. It's a "cute" way of downplaying how much you fangirl/fanboy/crush over something. You fangirl, therefore you are trash.
Whether or not you find that negative is up to you.
In the context of Dragon Age, and particularly Cullen, it came from a pretty negative place. This term started circulating around DA tumblrs between a certain circle of people who spent all the lead up to DAI hating on Cullen, and then started to like the character once the game came out and had to backpedal on the hate. The joke was along the lines of, well I spent all this time saying Cullen was trash but now I like him so I guess I'm trash too.
I don't actually mind the "trash" joking in general because it's usually a way of reclaiming insults with humor, but specifically in the context of Cullen... yeah, this backhanded side of his fandom who loves Cullen in DAI but must insist that he's so awful in DAO/DA2 so their hate was totally justified (see, all that "it's okay to hate DA2 Cullen because he hated himself too" nonsense, my personal biggest NOPE of this fandom) has kind of ruined the joke for me. I can't disassociate these "welp I'm officially Cullen trash now" posts from the implicit statement that you consider yourself trash for liking Cullen because you still think liking him is something to be embarrassed of, that being called a Cullenite is somehow an insult, that you're a fan of Cullen but not like "those" fans, etc.