Motivation for learning about fundamental groups and covering spaces

One motivation for an algebra seminar is to illustrate that there are good reasons to learn about groups. Many students may think that groups are just an abstract structure. However, groups arise at many other interesting places, like symmetry groups in geometry and physics, Galois groups in field theory and number theory, fundamental groups in topology and geometry, permutation groups in combinatorics and representation theory, and many other areas.
There are also specific good reasons to learn about fundamental groups. For example, certain compact manifolds are classified by their fundamental groups, namely compact Riemannian-flat manifolds. So the fundamental group "says everything" here.

From @DietrichBurde's answer, I'm assuming (too) that the original question is about the value of group theory to those interested in algebra, and indeed, the applications of group theory in mathematics he states are excellent.

My own approach to teaching math—and especially physics—at the undergraduate level is a bit different. I try to start with something the students already know, are already interested in, then show them what they don't know about this topic, and show how the discipline before them will empower them to answer those questions. In short, I try to exploit the students' innate curiosity more than some "career" usefulness.

As background: In my introductory optics classes I project a photograph of a beautiful double rainbow, which students all love and think they undertand and ask:

• "How does a rainbow arise?"
• "Why this order of colors (red on outside, blue on inside)?"
• "Why can't you ever get close to a rainbow?"
• "Why is a rainbow always this size, never bigger, never smaller?"
• "Why is the larger (secondary) rainbow farther out?"
• "Why are the colors in the secondary in the opposite order?"
• "Why is the width of the band in the secondary rainbow larger than in a primary one?"
• "Why is the broad range of sky between the rainbows ('Alexander's dark band') darker than the rest of the sky?"
• "Why does the rainbow move left or right when you do?"
• "What causes the pastel-colored rings just to the inside of the rainbow?"

And I do not answer these questions on the first class. I say, instead, "In this class we'll study optics so you can answer all these questions. Read Chapter 1 of your text. See you Monday."

So for group theory, I'd come to class with a Rubik's cube, and ask the following questions:

• "How many different configurations?"
• "Notice I can rotate the cube and the configuration is (effectively) unchanged. How many ways can I rotate and get an 'unchanged' cube? (We call these symmetries.)"
• "What is the 'most complicated' scrambling of the cube?
• "What scrambled state takes the most steps to 'fix'?"
• "If I perform this rotation (A) then that rotation (B), we get a different result from first performing B then A. Why?"

Then I would, wordless, merely hold up a $4 \times 4 \times 4$ Rubik's cube. Then a $5 \times 5 \times 5$ cube.

And I would not answer any of these questions (in the first class).

I'd end the class with: "The branch of mathematics that will enable you to answer these questions is group theory. Read Chapter 1 of your text. See you Monday."