For a bit of context, Schoenberg's atonal style was the culmination of centuries of development of Western harmony. From the Baroque/Classical era onward, the direction of Western music was towards greater chromaticism extending harmonies further and further away from the simple major and minor chords. Some composers came very close to complete atonality, particularly Scriabin and Stravinsky, but Schoenberg was the first to dive into completely atonal music.
While it may sound like someone randomly plonking on the keyboard, he had to work quite hard to get the atonal sound he wanted since harmonies are so ingrained in the way we hear music (especially for composers!). He developed the "twelve-tone system" in which each note had to be played exactly once in a theme so as to guarantee that no tonality was accidentally introduced by stressing particular notes.
The result is a style that is very cerebral, but enjoyable once you let go of the idea that music should contain traditional harmonies. Personally, I think the music world got a little obsessed about the idea and continued to develop it for maybe four decades longer than it should have. (His twelve tone technique was later extended to "serialism" which generalized the process to rhythms, dynamics, and instruments, and wasn't really abandoned by "serious" composers until maybe the 1980s.)
Incidentally, Schoenberg may have scared himself to death. He was intensely superstitious, particularly of the number 13. When he was 76 he was contacted by an astrologer who told him that this was a dangerous year since the digits of his age added up to 13. Later in the year, on Friday, July 13th, he spent the day extraordinarily anxious and died shortly before midnight.