There are a few big markets in the world where at least the top half of the customer base is under contract to the mobile operator, with some sort of handset subsidy (generally, this is really an interest-free loan) - mostly, the USA, Japan and Northern and Western Europe. Historically the subsidy was embedded in the contract price, and this made expensive phones look cheap. This was especially the case in the USA, where the contract price tended not to vary linearly with the price of the phone. Hence a $400 phone was 'free at $80/month' and so was a $100 phone, which had the effect of cutting the bottom off the market. This was a big benefit to Apple, which only sells phones priced in the top-half of the market, since it meant the discounted-to-$450 two-year-old model (at the time of writing, the iPhone 5S) was almost cheap as any (contract) phone on the market.
However, a big shift in the mobile business in the last few years has been to unbundle the 'subsidy' from the contract into a separate installment plan, making the price much more transparent to the user. There was a widespread narrative that this would damage the high-end of the market (partly Samsung but mostly Apple), but this doesn't seem to have happened at all.
However, shifting to an installment plan model does introduce a couple of interesting new dynamics.
First, unbundling the installments from the contract has often meant that the upfront cost of the phone goes away. Instead of paying (in the USA) $200 up front for a new iPhone, you pay zero. All things being equal, the monthly payments are therefore higher than the old subsidy that was embedded in the monthly phone contract, but it's now two separate fees, the service part is now cheaper and the increased monthly installment/subsidy element is directly linked to 'get a cool phone for zero up front', so it may be easier to swallow. This is one explanation for the apparent lack of impact on high-end sales of the move to installments - unbundling the installments and shifting to a zero up front cost model makes the phone itself look cheaper than before, even if the actual cost of ownership is unchanged.
Second, these installments have been accompanied by the introduction of a set of 'rapid upgrade' plans that effectively put the user on a permanent rolling contract in exchange for a new phone every year. You sign up to pay installments for some period longer than 12 months (it varies - 18, 20 or 24 months are in the market), but after 12 months you are given the option to get another new phone for 'free' if you extend the contract accordingly. To the consumer, the offer is simply that you continue making the same payments (roughly a dollar a day) but get a new phone entirely for 'free', which is pretty appealing. But that means you're effectively (if not legally) tied to your mobile network operator for the new, extended period - and then in a year, they do it again, and extend the payment again. So, your service plan starts at 24 months, runs out and then continues without a fixed term, but your installment plan keeps rolling on another 12 or 18 months into the future indefinitely, as long as you keep accepting new phones.
If you do the maths (and it's complicated and depends on your assumptions) these plans generally work out at roughly the same total cost of ownership as buying a new phone yourself for cash every year and selling the year-old one into the secondary market. There are a number of companies trying to handle that for you, but this makes it effortless. And for the operators, you're much less likely to churn off to another network.
So what happens to the old phones? When you take that upgrade, you have to hand in your old one. They go into the secondary market, which is rather the dark matter of the industry - we know it must be large and we can get some sense of that from survey data, but we don't have a solid number. One illuminating data point is the fact that for the last several years the number of iPhones that seem to be in China (if you look at data from companies like Baidu) has been rather larger than the number of iPhones that Apple's financial reporting implies could have been sold there. Second-hand closes some of the gap.
This takes us to Apple's entry into installment plans. Other than bundling AppleCare (that is, insuring your phone against smashing the screen) the plan looks pretty similar to those from the operators. The product page mentions that the loan is provided through CitizenOne, which suggest that it's not using Apple's balance sheet (though there may be ways to fudge this). But this brings a couple of changes to the market.
The obvious change is that if you're on, say, AT&T's Next plan, your next 'free' smartphone might be an iPhone or an Android - with Apple's plan it'll be an iPhone. So where for a mobile network these installment + rapid upgrade plans are a way to stop you churning off the network, for Apple they're a way to stop you churning from iPhone to Android. They're another way to build the resilience of the ecosystem.
The second change is that Apple is explicitly entering the secondary market. If you get a new base-model iPhone every year, you'll pay Apple $389 a year for a $650 retail-price phone every year. At the end of the year you give Apple the old one and it's going to have to do something to make that back, or more. Given the difference is $260 but the resale value of year-old iPhones (unrefurbished) today is $300-$400, that shouldn't be too hard. It does suggest more incentive for Apple to sell you a case, though.
The really golden conversion is someone who was buying a new phone every two years (and probably not selling the old one). Now, instead of $650 over 2 years for the base model, they pay $778 and the previous years' phones can be sold for another (say) 2x$350.
Meanwhile, this means Apple will be selling both refurbished models from one and two years ago and also (as it does today) newly manufactured instances of those older models. How will that work? How will the margins compare? Will it cost $260 to make an iPhone 6S in a year (maybe)? Then, how and where will Apple sell these? One obvious answer is that they'll go to emerging markets - to India, Latin America and, yes China. But how big will this be - how many people will take it up? How will Apple explain the difference?
This is also another piece in Apple's 'cheap iPhones by stealth' model. Rather than making an actual cheaper iPhone model (as it toyed with with the iPhone 5C, which wasn't actually cheap at all) - say in the $250-$350 range, Apple just carries on selling models from 1 and 2 years ago at lower prices. (Of course, this is effectively what all cheaper phones are - older technology.) This means it doesn't have to worry about price discrimination - an actual new mid-price iPhone could well have a larger effect on sales of the premium model than sales of the 'old, cheap one'. And people buy them. That said, this still leaves Apple with little to say below $200-$300, but perhaps it's happy with that.
Finally, this is another case of Apple moving into mobile operators' territory (which may be why it explicitly denied the recent rumor that it was planning to launch an MVNO - Apple almost never comments on rumors, but you only want to trench on one of your partners' prerogatives at a time). The immediate impact on mobile operators is actually pretty oblique - the rolling contract effect means that an iPhone user on Apple's plan won't be going into rival operators' stores looking for the best price for a new iPhone, so it has some of the same churn reduction benefits as if they'd taken the operators own installment plan. But this does mean that one more contact point - one more piece of surface area - has been removed. And the phones are unlocked. In the USA this is less of an issue since for a variety of (diminishing) technical reasons it's still hard to move a phone for one network to another, but in the rest of the world, which decided to use common standards, it's very easy. Hence, you can move from operator to operator in search of price or coverage independently of your handset. That's not necessarily a comfortable development.
Finally, this sort of shift is a prerequisite for there to be any point to soft SIMs in smartphones. A soft SIM in a subsidized handset locked to an operator is pointless (except for the engineering benefits). But a soft SIM (presuming the MNOs support it) in a phone financed by Apple, with seamless number portability (or of course a voice and messaging app environment in which mobile numbers matter less and less) is a much more subversive thing. The operators would of course have to enable soft SIMs (it's their code), but there's now a long history of operators bending to the inevitable.