As someone who has worked for both an insurance carrier and an insurance agent, the reason people buy insurance is two fold to spread risk out and to get the benefits (when applicable) of approaching risk as a group.

What you are really doing when you buy insurance is you are buying in to a large group of people who are sharing risk.  You put money in that will help people when they take a loss and in exchange get a promise of having your losses covered.  There is an administrative fee taken by the company that runs the group in order to cover their costs of doing business and their profits that they get for running the group well (or losses they take if they run it poorly.)

Some insurances are for profit, some are non-profit, all work on the principal of spreading risk around though and taking risk as a larger group.

So lets take a closer look at each of the advantage you get from participating in insurance.  The biggest and most obvious is the protection from catastrophic loss.  Yes, you could self-insure with a group size of one, by saving your money and having no overhead (other than your time and the time value of your money) but that has a cost in itself and also doesn't cover you against risk up front if you aren't already independently wealthy.  A run of bad luck could wipe you out entirely since you don't have a large group.  The same thing can still happen to insurance companies as well when the group as a whole takes major losses, but it's less likely to occur because there are more rare things that have to go wrong.  You pay an administrative overhead for the group to be run for you, but you have less exposure to your own risks in exchange for a small premium.

Another significant but less seen advantage is the benefit of being part of a large group.  Insurance companies represent a large group of people and lots of business, so they can get better rates on dealing with recovering from losses.  They can negotiate for better health care rates or better repair rates or cheaper replacement parts.  This can potentially save more than the administrative overhead and profit that they take off the top, even when compared to self-insuring.

There is an element of gambling to it, but there are also very real financial benefits to having predictable costs.  The value of that predictability (and the lesser need for liquid assets) is what makes insurance worth it for many people.