For at least some kinds of insurance, there's an economy of scale involved that reduces overall cost and hassle for some kinds of event. At many doctors and hospitals, you really want the negotiated healthcare rates that insurance companies get rather than the list rates. It's very nice if you are involved in a car accident to have an insurance company deal with the other parties rather than you having to do so.

For some, this may even extend to small amounts - it may be convenient to have replacement protection for your phone rather than having to agonize over whether to repair or replace if you drop it.