"So easy a caveman could do it" is repeated endlessly in these commercials, so that the slogan stays in your head. So the customer will think of that sentence every time they hear the name of the company. They also very effectively use humor in order to make the commercials, and therefore the message, memorable.
The underlying message is that switching from your insurance to theirs is very simple and easy to do. Most would think it's a pain with a lot of paperwork, so they hesitate. The commercial seeks to convince them otherwise.
Of course, the major message of this commercial (I’m thinking specifically of the airport one now) is that you should get Geico insurance. However, that is not ever explicitly stated in this ad.
As far as underlying messages go, there are a couple ways I think about this. On the one hand, you can say that we’re being taught that prejudice is hurtful and inappropriate. On the other hand, the ads are based on the idea that we should feel stupid if we are different. They are saying that if we don’t get Geico insurance we are as dumb as a caveman. If you look at it like that, the underlying message is that it is bad to be different.