My very limited understanding is that it is caused by food particles getting caught in depressions in the gut.
Close, but not the whole story. A
diverticulum is an abnormal pouch or depression in the wall of any body structure, they can occur in places other than the gut although diverticula of the gut are the most common to draw attention to themselves. Diverticulitis of the colon is inflammation of a diverticulum usually associated with bacterial overgrowth or growth in the diverticula of unusual bacteria. You can think of a diverticulum of the gut rather like a small meander or depression in a river bank, where things can accumulate in the lea of the river's flow and so also in the gut, the diverticulum doesn't get regularly scoured out by passing matter. There are a number of medical interventions that ought to be tried before resorting to major surgery, the obvious one being the one you've mentioned, mechanically altering what's flowing past by the addition of 'fibre' which can also beneficially alter the kind of bacteria growing in the vicinity too.
Diverticulitis isn't of itself necessarily a major medical problem. The two main problems with it are that if untreated it might in the longer term lead to further weakening of, and finally to a breach of the gut wall (and that is serious) and it is
excruciatingly painful. Pain from diverticulitis can be as bad as pain from bone cancer or a heart attack. The treatment of choice for major pain, opioids, can lower gut mobility and make the problem worse - double whammy. Saskia can be forgiven for any passing
"sense of humour failure"s in the next few days, I feel for her. Just writing about it I'm becoming conscious of feeling my own gut contents going about their normal movements.