Why it has been designed, where it has been used?
Its target market was terminals.
The 8275 is a very versatile logic chip for generating text displays (*1) so nearly any terminal format of that time, with up to 80 characters and whatever attributes needed, can be handled. It was made to cover the full range from simple 32x20 with 8x8 cells all the way to high end terminals with 80x25 using 16x12 characters.
It can generate any display with up to 80 characters per line and 60 character lines per frame with up to 16 (pixel) rows per character and arbitrary number of pixel columns, which allows quite great looking displays. This adds a lot of design freedom about how the display should look.
It's remarkable by having two line buffers to handle character timing independent from memory timing, while all access is done using external feeding. While Intel of course recommends their 8257 DMA controller, it can be done by any other logic as well. By using a 8275, integrated video can be designed without the need to synchronise system clock(s) with video timing (*2), as here video access synchronises with the host.
The other remarkable feature is that it didn't simply output some fixed size character matrix, but worked the data read more like a stream of instructions or a display list. Somewhat like Atari or Sinclair later did. A set high bit denoted a function, which could be either setting attributes for the character(s) following, or controlling memory access. For example a F1h code stopped DMA (and display) for the remaining line. A nice way to reduce memory load. Similarly, a F3h would stop DMA for the remaining screen. So a cleared screen with a prompt would only need to read some 5-6 bytes, reducing memory load greatly.
The whole 8275 was designed to not just create a display, but to be as flexible and minimal intrusive as possible — staying short of holding the whole screen memory local (*3). After all, even 'only' 4 KiB, for 80x25, on a single chip would have been lot in 1976.
While the resulting screen data may look odd from today's perspective, it was quite in line with how terminals worked — and allowed an efficient handling of memory and memory load.
Intel 8275 probably lived a very different life from its Russian brother KR580VG75.
Intel had quite a remarkable lineup of peripherals. Eventually the most comprehensive of all manufacturers. Where some just had very basic offerings, like some serial and parallel plus timers, Intel tried to cover as much high level functionality as possible within their family. There was not only the 'simple' brother 8276, but there were keyboard controllers (8278), even some combining keypad scan and LED (8279), GPIB (HPIB) functions (8291/92/93), HDLC (8273) and many more (*4).
Intel's designs were great and quite useful — but as so often simpler devices, using brute force and software instead of clever design, gave them a run for their money.
*1 - Well, in theory as well graphics, but then most of its features would be useless, so taking a less capable controller is more appropriate.
*2 - Within limits, as the controller should get enough memory access to fetch a whole line while the previous is displayed.
*3 - Like the 9918 did.
*4 - Heck, even including a DES chip, 8294. There was a DES hype in the late 1970s with Intel and TI (99541) supporting the lower end (400 Bytes/s throughput) for (credit card) terminals, while Fairchild (2002) and AMD (9518) targeted the high end (>1.5 MiB/s) host side. Motorola (6859) and Western Digital (2001) somewhat in the middle, usable for both sides.