Our common-sense notion of guilt tells us we should only feel guilt over a specific act that we commit, for which we take responsibility. The Brothers Karamazov works with a much more generalized notion of guilt: we are guilty for everyone, not just ourselves. That is, we are so intimately connected with each other that we are responsible for everything everyone does, including those who live on the other side of the world. The novel draws on the Biblical notion of original sin – that we are all born guilty because we've inherited Adam and Eve's primordial error in the Garden of Eden. The novel takes this notion of original sin not just as a fact (we're all stuck with it) but as a general principle that guides our actions. Since we're stuck with this guilt, we must act toward others with humility and respect, and we must share in their pain and suffering. The characters in the novel who arrogantly reject this sense of guilt are doomed to anxiety, isolation, and suicidal despair. In the rare moments when characters do recognize their guilt, they experience a sense of spiritual enlightenment and potential reconciliation with the rest of the world.
In Dostoevsky's Brothers Karamazov, the ability to feel guilt or shame is a necessary prerequisite to feeling love for other human beings.
Dmitri's trial is a satire on the legal notion of guilt, which ascribes responsibility for crime to one individual. In fact, as the novel shows, there are many individuals who contribute to a particular crime. True guilt should take a broader view of how the community as a whole contributed to a crime.