What's the interest for these breakout boards?
Q: Is there a market for the various break-out boards out there?

Yes, definitely. Some of the reasons:


Consistent little building blocks for rapid prototyping, and for beginners. One basically wires together a bunch of little boards, or better yet, plugs them into breadboards where feasible, rather than using a mix of PCBs, ICs and discrete parts
Prototyping with SMD parts where through-hole equivalents are simply not available. Not applicable to the examples in the question, but increasingly an issue as through hole parts get obsolete - for people unwilling  or not experienced with PCB etching or SMD soldering
Performance differences between through-hole and SMD parts - Not always, but often there will be some performance or rating differences, so it works best to stay with the package that a final product might be produced with. I've seen designs where the DIP IC worked great in a prototype, but the SMD part couldn't handle the heat.
Reuse: An IC survives only so many insert / extract cycles before the pins give out, unless using ZIF sockets (expensive!). For an experimenter, a beginner or children, this is a factor to be considered.
Convenience: 


A well-designed break-out board will often have the connections arranged in a more convenient sequence than the IC pin-out. 
Many break-outs will incorporate a decoupling capacitor right on the break-out board, saving some problems and some messing around. 
Labeled pin-outs, saving time compared to fudging around with an IC ensuring that the pins are connected right - important time-saver when "playing", or experimenting, when putting together and taking apart several circuit configurations a day. Saves time even for seasoned prototype makers.
Even simple things like LED + resistor for GPIO status indication can get old fast when using dozens across experiments, justifying, for example, one of the highest selling breakout boards I have been watching on eBay, a simple 6-LED indicator PCB:

This seller is making a killing on those boards, on sheer volume and a huge percentage margin.

Shipped cost: Not applicable to the seller mentioned in the question, they are pretty expensive anyway, but often, especially for non-US geographies, ordering a couple of units of an IC costs much more with shipping, than ordering a prebuilt break-out from some Chinese seller on eBay. This is first-hand experience for me in India.
Naivete of newcomers to electronics: Newbies actually pay for "rain sensor modules" which are essentially a set of parallel PCB tracks, and "analog temperature sensor modules" that are merely a through-hole thermistor on a PCB. Yes, the "convenience" and "reuse" points above perhaps apply to these too.
 




Q. Why are they priced unduly high?


You pay for the brand: SparkFun, AdaFruit and other well-known sellers often price their offerings significantly higher than the equivalent from no-name sellers. The argument that their products are better in manufacture rarely if ever holds true: More often than not the boards are fabbed in the same factories in China as the no-names. In some cases, the boards are identical and clearly from the same design and factory, even the "dust spec on transparency" flaws in the solder mask are identical. Brand and marketing, those are the differences.
Seller charges for inventory cost: High-volume breakout boards often sell at much lower premiums than the rarely-bought items. After all, the seller cannot fabricate small lots at a convenient price, so a minimum volume is built anyway, perhaps not fully populated, and then must be maintained in inventory (locally or at the factory in China) until sold. 
You pay for convenience: Point in example the LED PCB mentioned above, which despite the nearly $2 pricing, sells around a hundred pieces each day in each color, from one seller alone. 
Seller charges for buyer's constraints: An individual can typically not get the same kind of price, quality guarantees, and turn-around cycles from PCB fab shops, that a high-volume seller can. Since these sellers have the volumes and pipelines and solid contracts in place anyway, it's trivial to add a few not-really-important break-out boards to the order queue, sometimes merely as fillers in a PCB panel order. 




Q: How are some breakout boards cheaper than the price of the IC on Digikey and similar sites?

Key example: The Analog Devices AD9850 DDS boards sell for as low as $2 each on eBay if you watch for it. The IC isn't even available at that price.
   


Volume purchase discounts. On sites like DigiKey, it is educational to examine the discounts for a given IC in 1kU and 10kU quantities, compared to singles. 
Geography: Many manufacturers of ICs and discretes sell direct container-load shipments far far cheaper than any DigiKey or Farnell price point you will find, if the destination is in the same country as the semiconductor fab facility, or even a nearby country. With the rise in semiconductor fabs in China, Malaysia and other parts of the far East, this means a price edge in that region.
Shipping cost per unit, retail versus wholesale: This one is basic economics. 
"Fallen off the truck": Shipments ICs and even complete assembled modules / breakout boards such as bulk orders from SparkFun get "misplaced" in transit, and then get sold on various sites by nameless sellers where the provenance of the product is non-existent.
"Dark shift" manufacture: PCB assembly factories and semiconductor fabs have been known to run an extra shift producing the same item they are contracted with some name-brand customer to produce, in an off-shift, or by misreporting production runs or failure rates. These additional units are then sold at whatever the retail market will pay, evidently cheaper than the brand-owner's pricing. 
Factory rejects: Manufacturers often have very stringent rejection criteria for assembled boards, or for ICs. The rejected units may well be functional, just not at the maximum performance ratings. These items are officially "destroyed" at the fabricator, but in reality often find their way to the grey market.