From what I can tell in my daily life (as well as scrolling through music forums), atonal music, pieces that do not revolve around a tonal key as we are used to, is a rather controversial subject in the music community. (Here’s a list of atonal composers and compositions).
You might love it, tolerate it, or hate it, though most people seem inclined towards one of the extremes – (usually the latter). I have good friends on either side of the love-hate spectrum.
People who do not like atonal music usually feel that it is mostly a mesh of random sounds that has way too dissonance (dissonant notes are notes that do not fit the current harmony/chord that typically causes a feeling of being unresolved). While it is not my favorite type of music, I find this claim uncalled for. The strange, “wrong” notes we often hear are not necessarily random, and dissonance doesn’t really hold the same meaning in this context.
There are many claims as to why people generally dislike atonal music. Perhaps a grounding on tonality is “wired” into our brains. Maybe because we are often influenced and raised with very consonant, tonal music from birth, the sounds in atonal works sound wrong or strange. This could just be a case where most people hate it while the trend is growing, and it will be celebrated more as time passes. I have yet to find a definitive explanation for this situation.
One thing of note is that film scores frequently incorporate atonal passages, which evokes tension. When accompanied by the film, the overall quality is magnified. The audio alone, however, tends to provoke the same rejection from most people. There is certainly purpose and meaning to atonal music that is underestimated.
I think one way to learn to appreciate atonal music would be to look at the history of the piece in question. It might give you a clue as to what to listen for. A piece of atonal music that I enjoy and play is Muczynski’s Toccata Op. 15. The appeal to the piece lies in the composer’s manipulation of the rhythm to create a rather “angry” effect. In fact, this is a piece Muczynski wrote as a “rage” piece after a car accident. He likened it to Beethoven’s “Rage Over a Lost Penny,” calling it his “Rage Over a Lost Car” (See page 154 of this dissertation). Knowing this bit of history makes it a lot more fun to play and listen to.
I know composers who absolutely love atonal music, more than tonal music even, because there is more freedom to try new things or implement interesting effects into their compositions. Compositions teachers teach and encourage the use of atonal techniques because it often makes the resulting works more interesting and fresh, giving the students more to work with.
People who love music should give atonal music a chance. At the very least, you can broaden your horizons. And perhaps you’ll find a new love and appreciation.