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In psychology, as in other social sciences, you have different "models." People often say that depression is a chemical thing, but there's also a cognitive-behavioral model that sees depression as a failure of what they call self-regulation -- the ability to control one's behavior and maintain mental states, such as concentration. For those who think in terms of chemistry, it's natural to say that bad chemistry and bad mood results in incorrect thoughts, which is true, but it's also true to say that false thoughts can result in bad moods and bad chemistry.

I can't completely explain the delusion, but I can show you how it can happen. When someone has high anxiety there's a tendency to have fearful thoughts, possibly paranoid ones. Those thoughts can increase the person's anxiety, with a vicious circle folowing -- more negative thoughts, etc. In a situation like this what the person should have, aside from medication, is stress reduction and some calm discussion of the things that are bothering him. In cases where the person may do something harmful, there's often inpatient treatment.