The purpose of infomercials is to sell products, of course. But the way infomercials sell products is not just to present the product but to get the viewer emotionally attached to it. You don't buy GLH-9 spray-on hair because you're bald; you buy it because going bald means you're getting old and probably impotent, and the product will stop people from noticing. You don't buy the Showtime Rotisserie because it can hold a ten-pound turkey; you buy it because you want the luscious, mouth-watering baby back ribs and tempting, juicy, golden-brown roast chicken that are going to magically spring forth from it.
But there's one thing that will break the mood for almost any viewer: a high price tag. Those baby back ribs might be delectable, but are they $160 delectable? Not likely.
So what's an advertiser to do? He can't outright lie about the price; that's illegal. He could fail to mention the price, but the customer will find out anyway if and when he actually orders the product. But what he can do is reduce the emotional impact of the price by dividing it by three or four. Suddenly he's not asking for $160 of your hard-earned money, all at once; he's asking for "four easy monthly payments of just $39.99".
It works because people fall for repetition. They hear the $39.99 over and over again and convince themselves that they're being sold something for $39.99, which sounds great: after all, $39.99 isn't really that much—less than dinner and a movie for two. It never even occurs to them that they're actually spending $160, a fair hunk of change. The constant repetition of the words "easy" and "just" also reassure the reader that he can easily afford the product.
May be preceded by the presenter telling his audience the price they aren't going to pay for the item, usually referring to the "fancy department store brand" or "the price you thought you'd have to pay for this". Even after revealing the Four Equal Payments, the presenter may shock the audience further by removing one of the payments entirely. That's right, you only need to make Three Equal Payments!
In British infomercials, the full price is usually mentioned as an option—you can pay the full amount or you can go for the payments (but see below). In North America, though, the full price may only be given at the end of the infomercial by the announcer who tells you that Operators Are Standing By, or even rushed through in the tiniest of mouse-sized legal type which is sneakily asserted as a payment option only available to those who send a check or money order through the mail, which these days is a quickly declining amount of people.
More egregiously, shipping and handling will usually only be mentioned in passing even though shipping and handling are often quite high on these items, especially in North America—and it's usually charged in total at the time of the first payment. Cue the legions of dissatisfied customers whose first "easy monthly payment of $39.99" was well over a hundred and twenty dollars, because shipping and handling was eighty bucks.
In the UK the phrase is "four equal payments", which doesn't sound anywhere near as sexy as "four easy monthly payments"—and that's deliberate, as the word "equal" is mandated by law and is expressly meant to remind viewers that they're paying in instalments. However, the UK has an additional, much sneakier variant: the amazing product that's sold for only £99.99, or 12 convenient equal payments of £10.99, plus shipping and handling. Do the math: 12 times £10.99 is £131.88, or over thirty pounds more than the purchase price for anyone who paid for it at one go. Basically you're paying about 30% interest in order to not pay it all at once. note This trick is also done with rent-to-own sales, where the total payments can end up being two to three times the original value of the item being purchased. The UK does at least have the advantage of the stated price always including tax, unlike the US where that is added on top.
A similar tactic is often used by furniture stores where payments are deferred: "Do not pay anything, not even the taxes, until 20XX!" On a more serious note, sub-prime mortgages operated on the same kind of idea on a much larger scale.
Related to the other ad gimmick Just Pennies a Day, which is this taken to a much larger scale.