Manuscripts are the future gold for book publishers. at every step of the book-publishing industry there is a slush pile full of manuscripts. Most of them will never be published, some have potential but need to be edited and some will be snapped up by agents and publishers. Reading manuscripts for a living can be achieved at several points along the way.
Become a literary agent. This entails reading manuscripts during about 90% of the time. Manuscripts are sent to your office, they are emailed and probably slipped onto your doorstep as well. Everyone wants to get their book into print and it's your job to get a book deal for the best of those manuscripts.
Work as a literary agent's assistant. If you don't have the experience and connections to become an agent, become the assistant of a literary agent. As the assistant, your job will be to go through the slush pile nonstop, reading manuscripts all day, every day, and recommending the best of those to the agent.
Go to work at a book publisher. As an editor or an editor's assistant, you will have a slush pile at the publishing house to go through. Some publishing houses allow only manuscripts that have gone through a literary agent first. Other book publishers allow manuscripts to come in from anyone. Those publishers receive enormous numbers of manuscripts, all of which should be at least skimmed to find one worthy of being published.
Be familiar with what sells. Reading manuscripts for a living, no matter what the job title, depends on knowing what kinds of manuscripts will sell and make a profit for the agent or the publisher. That usually requires an extensive study of the book market and constantly keeping on top of the new trends in the book publishing industry.