What is the use for built-in graphic card on a "gaming" motherboard?
There's a few. Firstly, nearly every modern single mainstream1 processor has a integrated on die GPU. The chipset supports it. Essentially your only cost is the traces and connectors, so it's a 'free' feature you can design in - unlike older designs. Interestingly, many of the Sandy and Ivy Bridge-era Intel chipsets outside the Z series made you pick one or the other (H series) or didn't have onboard video (P series). Many earlier processor families used a PCIe 'slot' for a onboard chip but most integrated graphics is on die.
Modern integrated GPUs do neat stuff like quicksync, which mean even with a discrete card the IGPU bit of your core can be working. With earlier drivers you needed a display (or a dummy display) but you can set up quicksync to work without one for faster transcodes or video playback. I'm sure AMD has something similar on their APUs - but I've not used them recently - they're somewhat more powerful than intel's models, and paired with a discrete radeon might do switchable graphics to save power.
It's also handy if your main video card's blown and you don't have a spare. I seriously found this useful with my last PC, which had GPU failure. Sure, you can replace it, but its totally worth it to be able to check just by yanking out the old card and changing the output the monitor is plugged into.
So, in short: "All the pricy stuff is already there and Intel insists, so why not add a cheap feature?"
1 I'd consider most Intel LGA 115x processors, and AMD APUs to be mainstream. The AMD FX series and Intel LGA 2011 are enthusiast focused, though the FX series kinda overlaps with intel's mainstream products on price. AMD fans may disagree.
As of 2018 - things get a bit more complex. Intel's core i3 and i5s are solidly mainstream. The i7 and i9 badges have mainstream and server inspired models. As for AMD - ryzen's the mainstream and threadripper is enthusiast.
There are few wrong assumptions and they led you to wrong conclusions:
Many motherboards marketed as "gaming" has an integrated Intel graphic cards.
The graphic card is on CPU. Intel made this decision, not motherboard maker. When buying Intel, GPU cannot be avoided.
I understand that putting in an integrated graphics into a motherboard increases it's cost.
It depends on what you're looking at. If you look at price of chips alone, the costs are not ground breaking. On LGA775 platform the GPU was integrated in chipsets, so some had integrated GPU while others were genuinely lacking the processing power. However, the low-end chipsets with GPU (eg. G41) were actually cheaper than high-end chipsets without GPU (eg P45). So we can conclude that while integrated card must increase the price of a chip, it's not really enough to justify costs of making 2 lines of chips: with and without. This is probably why Intel decided to put a GPU on every single consumer CPU.
Now, since the GPU is already on the silicon, we can consider the costs that can be decided by motherboard designer. If he wants to make the GPU work, he adds the connectors (probably the most expensive part of implementing onboard GPU this days), traces, and a handful of dirt-cheap passive components like those tiny resistors and capacitors. Those costs are still negligible. If we were talking about lowest-end budget motherboard, axing few dollars would probably be at least put under serious consideration - but on a high end motherboard that is already expensive any possible savings are negligible.
This is because integrated video do not have a chance to compare with higher end Nvidia/Radeon offerings.
I cannot really call you wrong on this one. With high-end they can't compare. However, the old wives tale that integrated GPUs are useless isn't true anymore! There are 2 desktop Intel processors (LGA1150 Broadwell, Core i5-5675C and Core i7-5775C) that have integrated Iris Pro Graphics 6200 that was a shock when it was released in Q2 2015. It's performance is comparable to low end discrete GPU, so it can be actually used to play most games on lower detail. If you're a gamer on tight power or space budget (eg. console-sized living room PC), I believe this would be a way to go. This integrated GPU was probably quite expensive, that's why it's seen only on $276 CPU.
There is also an elephant in the room here. I believe you've assumed that "gaming" means "top performance". Well, it does not. It's simply a marketing strategy. Nobody is really able to tell what "gaming" label means, except that it features aggressive styling and a higher price tag. Basically a premium product. So, when in doubt, just add every feature you can and you'll have one more point on feature list. Like pretty RGB lights that most users will probably lock up in the case and shove under the desk to be never seen again or shiny metal over PCI slot that does nothing but looks cool. (Seriously, lights? How are they in ANY way useful in gaming? I can't believe you questioned integrated GPU while there are lights on the mobo!)
Could you please explain what it can be used for (I'm guessing the intentional use, but any other possible uses too) given that for a gaming PC one is most likely to utilize an external GPU?
There are two uses I can think of for integrated video in enthusiast hardware:
It can drive an additional monitor. Have one or two monitors driven by the expensive PCIe GPU. Use those for your games that demand performance. Drive an extra monitor off the integrated video and use that for email and web browsing. Modern high-end cards tend to be able to drive more and more monitors, but I still think this is a valid point.
My primary system has a monitor I use to keep up a web browser while gaming. It is great for having a wiki, forum, or Arqade up with information I can use in the game. My GPU has enough outputs that I can drive it off the primary video. If it did not, I would not hesitate to plug it into the integrated video connector.
Troubleshooting if your GPU is damaged. You still have backup video to use your system while troubleshooting your primary video, or while ordering a replacement and waiting for it to arrive.