Insurances protect you in the case of damage because the person who caused the damage has to pay the costs. Private liability insurances (Privathaftpflichtversicherung) are useful because in the case of damage the costs can be very high. The insurance will take the costs for damages which are not caused deliberately. For example, if you dropped a camera of a friend or somebody injures himself/herself by a flower pot which has fallen from your balcony.
Furthermore, household goods insurances (Hausratversicherung) are one of the most important insurances but it is only worth it if your furnishings, clothes and other goods have a high value.
Usually contributions for insurances, radio and the gas and water supply are debited automatically from your bank account. Your bank account must be covered on that date; otherwise the bank charges additional bank fees. If the insurance cannot debit the first amount due, you are not insured.