The first note here is that they all say that "people who switched" saved. Kind of obvious, but people don't usually switch to a higher or equivalent insurance premium. Everyone saves when they switch. And it makes little difference which company people switch from or to. They won't do it unless they're saving money.
My second thought is that the numbers are always set in annual savings. Most people (by which I mean me at least) don't really think too much about their premiums in annual terms. They pay (and think about it) either monthly, or every six months. So when comparing the supposed average savings to a person's six month premium, they need to cut the advertised savings in half.
The third reason these numbers are so high is that most people that don't have lots of accidents or tickets already have pretty low rates, and don't want to spend the time and effort to analyze and compare the different numbers and contexts for insurance when their possible savings will only be a few dollars per month. Maybe people should, but I don't think they usually do.
Who does look around and switch policies? People who are paying a lot of money for their insurance. Young people (don't get me started about yound whipper snappers...) Accident prone people (in the automotive sense of the word.) Illegals (by which I mean people that get lots of tickets for speeding, illegal lane changes and U-turns, driving while under the influence, etc.) The poor (people with poor credit, owe insurance companies money, etc.)
Most of these people have high premiums. And I think almost all of them are acutely aware of when changes in life circumstances, or time, will reduce their premiums. If a person's been paying an insurance company a ton of money, and they think they can get a lower premium, I think most people would call other insurance companies first. And then they save a ton of money. This is where those huge average annual savings come from.
All of this adds up to the insurance companies finding their average savings to be a very good marketing tool. The problem is that every insurance company can offer a similar line "Our company saves the average person who switched to us X hundred dollars." They can even be specific. And that's why new Geico customers save hundreds switching from Progressive, and new Progressive customers save hundreds switching from Geico.
It may be a completely true number, but if every company can do it then you're marketing to those that are too ignorant to realize that the number is virtually meaningless. My wife and I called Geico, All State, and Progressive some months ago, and premiums that covered less from them would have cost around $10.00 more per month. I'm guessing they have to charge more to cover their massive marketing budgets. My company doesn't do a lot of marketing, and it's better insurance for less money.