The general idea of insurance, in my opinion, is to cover disasters. Most people can't afford to keep enough money in reserve for major, unexpected, and unavoidable expenses. If you have $X in your savings account, it's presumably because you need that money to cover your regular living expenses. If you have a catastophe that requires you to spend $X to fix the house, what will you use to buy food once you depleted your savings? Insurance allows you to continue living on your savings and earnings.
And if you actually do have enough money to cover such an event, you're wasting it by keeping it in liquid reserves. You can do much better by investing it. If the investments are easy to sell (stocks, mutual funds) you can dip into it to cover emergency expenses, but this would mean you have to sell at inopportune times, maybe without having time to research the best options, and you'll likely incur capital gains taxes as a result. Self-insuring like this is not really a good option for most people.
A good example is medical insurance. In typical years, your medical expenses might be a few hundred dollars for routine doctor visits and medicine. For a family or a person with some chronic conditions this might reach into the thousands. These may be affordable to pay out of pocket, so you might wonder why you're paying several times more than that for insurance.
But medical costs can be very unpredictable. You could have a heart attack or get cancer, and the costs of treating these are likely to be in the hundreds of thousands. Unless you're rich, this can easily bankrupt you. Maybe if you're lucky, your savings from not paying all those insurance premiums over the years would cover these expenses, but that's hard to say. Can you be so disciplined that you don't spend those savings on frivolous things like vacations and more "stuff"? So in this respect, buying insurance is kind of like putting money in a retirement account -- you're sure the money will be there if you need it. Although unlike a retirement account, if you're lucky enough to never be severely ill, you may come out behind financially.
Of course, the above doesn't apply to places with socialized medicine, and is the major reason for socialized medicine. But the general idea applies to any insurance, although the amounts tend to be lower. This will affect the necessity of paying for insurance. Most people advise against buying extended warranties for small electronics and appliances, because the replacement cost is relatively affordable. But if your home is detroyed in a fire, that would be a disaster comparable to a heart attack (if you have a mortgage, they may even require you to have insurance, because they don't want you to abandon the home and leave them with the worthless husk).
If you have decent savings, you can use it to make a compromise. Most insurance plans have a "deductible" -- insurance doesn't start paying in full until after you've spent that amount out of pocket in a particular year. You can get lower premiums by agreeing to a higher deductible.