Why do things fall apart?
The answer to this question depends to some extent on how deeply you want to delve into the problems and questions raised by the novel. On the surface level, it is clear that the hard-headed and stubborn Okonkwo is headed for some kind of fall, he fits well within the bounds of the tragic hero, with some fatal flaws but plenty of strengths, particularly those involving hard work and combat.
But the entrance of the white men and the christians to the village and the country in general create conflicts between the traditions of the villagers and the beliefs of the new visitors. These conflicts are not going to be resolved simply or easily and Okonkwo's stubbornness perhaps leads him to be more directly and violently opposed to them.
In the end, I would say that it is the inability of either side to communicate effectively with the other that leads to things falling apart. Whether it is within the plot of the story itself or the deeper ideas brought up, the two sides cannot come to any kind of meaningful dialogue and so, as you noted, things fall apart.