When I introduce groups I first go over a very (e.g., 15 examples) long list of particular groups. I go over each example and verify the group axioms without naming it such. Then I present the problem of studying all of these examples together, motivated by simply saving labour. Instead of proving things again and again for each case, we'd like a single uniform formalism in which to study all of them. From there the road to the definition of abstract group is quite short.
A bit further into the course, when we get to polynomials, I explain that polynomials are an excellent algebraic way to encode combinatorics and therefore it is important in lots of places (since counting is fundamental). I present the students with the question "Can you design two (biased) six-sided dice such that the probability for each sum between $2$ and $12$ is equal". The negative answer can be obtained by considering the generating polynomials.
Of course, if such theorems as the Burnside Lemma in group theory is covered, then lots of real-life applications can be given. I also like to very briefly mention some of the applications of group actions in chemistry.
I also find that many students try to visualize what an abstract group is. This is impossible of course, and for students who never took any abstract course, and are so used to just computing or drawing graphs, this is a big psychological hurdle. I strongly advise, several times, not to even try to visualize. Rather, they need to understand that they need to develop a feel for the axioms and the proofs, of course, through practice. But don't visualize!