There were a lot of reasons. And, of course, not every composer had the same reasons or beliefs or practices.
One reason is ideological: with the rise of musicology as an academic field in the early 20th century, scholars desired a linear narrative for telling the history of music. According to this narrative, tonality reached its peak during the classic era (Mozart, et al); then romantic composers like Chopin and Liszt began experimenting with stretching its limits; and then late-romantic and impressionist composers began breaking the barriers altogether; and by the early 20th century, tonality had died. So atonality was seen by certain very influential scholars and composers as the natural next stage in the history of music. Of course, this ideology is really baloney, because tonality never died, and the 20th century saw an explosion of tremendous creativity with developing new tonal styles of music. But this is the ideology that guided how music history has been taught for 100 years.
Another reason is, well, also ideological. Not only did some people think that tonality was no longer fashionable or modern; it was entirely used up. Tonality had been fully explored to all of its limits, and nothing new could be done with it. So atonality arose as a totally new paradigm for doing new things. Again, that's baloney, because there are so many new styles of tonal music today, but again, ideology is ideology and influenced the way that music history has been taught.
Another reason is, again, ideological. Tonality favors certain notes more than others. The tonic and dominant, for example, are given much more significance in tonal styles than, say, the subdominant or the supertonic. By the same token, certain kinds of chord progressions were given more significance than others. One of Schoenberg's goals was to create a sort of tonal egalitarianism, by stripping all notes of their hierarchical values. Why should the tonic be any more important than the supertonic? Why should consonant notes be valued over dissonant ones? So part of the logic of the strict 12-tone system (which was only one kind of atonality) was to make all notes entirely equal, by stipulating that no single note can be repeated until all 11 others have been used as well.
These are a few reasons. :-P