There are three main reasons why backing claims up are important on a site about literature.
The first reason is that backing claims up is crucial to deter false information. Fake news is a huge problem on the internet because anyone can write falsehoods anonymously on the internet. Although the media attention has focused on hostile actors deliberately spreading false information, a lot of false information online comes from well meaning people who simply misremember a critical fact or misunderstand a central argument. Backing claims up solves this: it makes it very easy for a reader to go back, retrace your work, and make sure your answer is correct (and leave a comment if it isn't). If an answer isn't backed up, verifying claims and telling false information from truth becauses infinitely more difficult.
The second reason is that students of literature, and people conducting rigorous literary research, are one of our site's main audiences (although they are not our only audience). If you think about why someone would google "what does Moby Dick symbolize", it should be obvious that students would be at the top of the list. If you think about who would be likely to write expert questions and answers, it should be obvious that someone who had just written, say, a paper would be at the top of that list. As the study of literature is based on the art and practice of making compelling arguments, answers that don't back their claims up are of little use to one of our site's main target audiences. Imagine you're writing an essay on a book. An answer that quotes or otherwise references a specific passage of that book will be much more helpful than an answer that doesn't.
Finally, the most important reason is that unlike many stack exchange sites, the Literature Stack Exchange is about a subjective topic: literature. There are a lot of questions on this site, such as Do Guildenstern and Rosencrantz deserve to die?, where the correct answer is often a matter of opinion. For such questions, the only way to make them work in Stack Exchange's Q&A structure is if answers back their opinions up with arguments. If these questions are seen as a place for anyone to post their unsupported opinions, then the question just turns into a poll where people vote for the answers they personally agree with. Polls do not work on Stack Exchange. But if people write answers where they explain why they believe in their opinion, then answers can be evaluated on the basis of who has the most compelling explanation, and a genuine sharing of perspectives can occur.
Related and useful reading:
Good Subjective, Bad Subjective
The original Moms4Moms Back It Up! rule