As far as abstract algebra goes, I think you should make the analogy with your students that when they learned the four basic operations as a child that they did not at the time have full appreciation of what they were learning. To explain to a seven year old the "application" of balancing a checkbook or calculating the tip on a check would have little meaning to them. Similarly, when learning about groups or rings for the first time, they are entering a whole new world of calculations that they can perform. At first, they will learn the rules without a full understanding of what they are doing. In due time (i.e. later in the course), after much hard work on their part, their understanding will have increased and they will be ready to appreaciate the applications of their new skills. I generally find students are willing to give you some slack, at least for a while, but you do need to make sure you fulfill your promise so that the students do get at least a glimpse of some interesting applications before the course is done.

As far as specific applications go, using Burnside's Theorem for enumeration problems is a fun application of finite groups that students enjoy once they "get" it. It is also worthwhile, assuming the students have a linear algebra course in their background, to go back and reinterpret diaganolization and Jordan normal form in terms of group actions. It is criminal how many math majors get a degree without having any concept of why diagonalization and Jordan normal form matter, despite how important eigenvalue-eigenvector analysis is in real world science and engineering applications. It should be possible to introduce the basic idea of Klein's Erlangen program (in the familiar example of Euclidean geometry) for students to see another example of symmetry at work, though this might be a bit ambitious. I would avoid problems such as classifying finite simple groups of order $< n$ (your choice of $n$). I think that the few students who actually are stimulated by such problems are the ones who generally need the most encouragement to unleash their creative side to complement their appreciation for logical rigor.