I'm currently finishing up my MSLIS. These are reasons I seemed to see are as follows.
Some people think that the library's only librarian is found at the reference desk (or perhaps one at reference, one in children's). They assume the rest of the work is done by people without specialized schooling. This is not the case, though there is a worrying tendency amongst today's libraries to hire outside the pool of MLS grads. In some cases, this is normal--for instance, at the library I worked at in high school, the circulation desk was not manned by librarians, because the work was largely rote. However, a lot more people see the circulation desk than see, say, technical services, so they get the impression that the people at circulation are the only people in the library.
Going along with this is the fact that many people think that what librarians do is not very complicated. This is both true and false. Some of what librarians do isn't complicated. A lot of it, however, is more complicated than it looks. For instance, what seems to be a simple children's program actually requires a lot of thought and coordination; it's not unlike some aspects of an elementary school teacher's job. People in LIS programs aren't just learning the Dewey Decimal System; they're learning about all the different things a library has to do to function as a member of a larger community. They focus on community engagement, and emerging technologies, and problems of access, and a bunch of other things. These things can sometimes be critical:
As an example: your library is running out room because both the laws of time and space and your budget are completely inflexible. What do you do? Stop buying new books? Cull your existing collection? If so, using what criteria. Recently, a local library decided to cull their collection simply by discarding books that hadn't been checked out in a year. This resulted in all of the books on some subjects being discarded, which was Not A Good Thing and got some media attention. A good LIS program will teach students how to approach these problems.
Additionally, a lot of people assume that an LIS degree only qualifies you to work in a public library. They forget about corporate libraries, legal libraries (which often require a JD as well), museums, and academic libraries. They also forget about the IS part of LIS, so they don't think about things like web archives, data management, digital collections, data analytics (some places offer this), and other not-library places. This is my particular frustration; people will ask me about, say, library-centric taxonomies, and I have to tell them that I don't know, but I'd be happy to manage their database/do semantic analysis/prepare metadata on digital objects.
Finally--and this is potentially the most controversial one--librarianship is seen in a feminized light. As a result, it suffers from the same sort of devaluation that teachers and (increasingly less frequently, due to a crisis in numbers) nurses have dealt with for decades. Gender roles aside, it's also frequently seen as a labor of love, something that's romanticized and that everyone who really loves reading is somehow immediately qualified for. And, you know, since you love it, you obviously don't need special training or decent pay, right? Going to school for it seems to be viewed as somehow indulgent and wasteful as a result.
All of that aside, there do exist people whose entire thought process is as follows:
"I have/will be getting a XXX degree. It's not that useful. But I really love reading and books. I know; I'll go to library school! Then I can read books and be around books all day and people will pay me and it will be awesome!" I was that person when I left undergrad, and it would've been a terrible idea for me to go for my MSLIS at that point. All I would've been doing is taking that option because I didn't know what else to do. That's a terrible reason to go to graduate school. That applies to all fields, mind, but the lack of direction is especially obvious when it occurs in this case.
A love of books is great--and most people seem to have that; I mean, I'm not even in the "book" part of LIS, and I love books--but that's about 1/100th of what you need. The people who seem to really thrive in the LIS program here are the ones who are driven to engage on a community level, the ones who not only embrace but spearhead change, and the ones who have library experience and are looking to deepen what they're already doing at work. The people who don't fit those criteria...they don't seem to work out as well.