How important is it to show students an application of the topics seen in an undergraduate course?
I can't point to any specific research, but it is a common observation that (i) students have difficulty putting individual topics learned in class into a context that would allow them to see where they are used, (ii) students who don't understand why a topic is relevant tend to forget it rather quickly. This is as true in mathematics as in many other areas, and I think it is widely thought that this limits the ability of students to apply what they learned to new situations.

Consequently, at least to me, it is of great importance to provide students with the context in which each topic I cover in my classes lives. Why do we care about a particular statement? What implications does a particular theorem have on areas they are already familiar? I know that this approach is not universally shared in mathematics (or at least others may define the context and the applications to be rather narrowly as "related areas of mathematics"), but at least in my view, it is important to show students what the material they learn can be used for. Otherwise, mathematics is no more than an idle game of moving symbols from one side of an equation to another.