An audio engineer is like a guy who goes and fixes up an older car until its fully restored, and then spends time detailing, cleaning and painting it until it is 100% blemish free. In mixing we often change many functional aspects of sound and how we hear them, sometimes a new mix can make songs sound brand new. If the music itself is the engine, you can think of mixing as getting any dents out of the car body, as well as sanding and buffing until smooth. Mastering is the last details, the paint, the wax coat.
Mixing and mastering are arguably some of the most important skills starting musicians can learn. It is a direct part of producing music, and producers who can mix and master have much better sounding beats. Personally, it drove me crazy listening to my early beats knowing they were nowhere near mixed, and I had to learn how to finish them all the way.