How do adult teeth get cavities?

Neglect mostly: People get cavities mostly because their mouths have a prevalence of cavity causing germs (strep mutans/lactobacillus sp.). There are many other factors to consider: saliva, diet, ph, and others influence your overall cavity risk. The factors you can control to reduce risk are diet and home care. Reduce consumption of sugary/acidic foods/bevs. Brush/floss daily, and chew sugarless gum w/ xylitol.

New and recurrant: Adults get decay the same way kids do: the interaction of sugar, oral bacteria, tooth, and time. If you already have dental work, it can leak or break and cause recurrant decay. In older adults, where salivary flow is very diminished, that can also encourage decay. Many drugs, prescription or self-prescribed, can also decrease salivary flow and thus encourage decay.

Bacteria: A cavity is an infection that has damaged the hard tissues of the tooth, and can spread into the soft "nerve" of the tooth and from there to the rest of the body. Mostly cavities are due to not cleaning between teeth (floss) or improper brushing. Sometimes exacerbated by other factors such as reduced salivary flow, old worn out restorations, fractures, etc.