When libraries offer RSS they make it possible for people who are invested in their local library to have alerts sent to their personal feed reader instead of having to visit the library website continuously and search through the various pages that a library website might have. This is a useful way for libraries to alert their patrons to new book arrivals as well as to special events and notifications that library patrons might be interested in.
The Library Channel at Arizona State University Libraries (ASU) offers a RSS feed that is continuously updated and contains not only information pertinent to the library but also to the ASU campus. The library also uses this feed to promote the Library Minute videos that the library films.
The Tacoma Public Library goes a step further with their RSS feed by offering different types of feed to their patrons. The feeds they offer are: library events, new arrivals, and new arrivals by category. Along with a list of feeds the library also explains what RSS is and gives some suggestions of popular ones to install.
RSS allows a library to extend its online presence, and as more people have, and continue to be used to finding their information online this is too good of an opportunity for libraries to ignore.