I work at one of the relatively few studios that retain a full-time writing team. I also work very closely with the writers in my role as a tools developer. We have around seven (I think) full-time writers, including their lead, and two writer/designers who act as lore directors and generally are the final authority on the direction of the world.
For the most part, the writing team authors new text or edits existing produced by other designers. The kind of content they generate tends towards the minor characters, dialog scenes and conversation, bits of chatter and background VO, quest description, descriptions for areas -- basically, anything that needs to be written. They also work with those designers to help ensure that all the text that is produced fits within the framework of the universe we are creating -- that nobody uses anachronisms, that characters of a particular cultural background exhibit the traits of that background, that the game is not too chock-full of pop culture references, et cetera.
Our writers also work with external partners, such as the authors of books based in our world, to ensure accuracy and adherence to what is canon. Most studios maintain a "story bible" of some sort that documents these sorts of things, and maintains guidelines for what is and is not canon (ours are not public, but Bungie has published this guide to what is considered canon in Halo, for example)
If you have a strong background in literature, writing, or related fields you may be able to break in to the industry provided you can find a company that employs a team of full-time writers. This may be difficult. I would caution you that is far more about editing and creating in-the-small than it is sitting around and getting to dictate the entire plot of a game.