Because We Can!
We can cram all the most important components of a computer into a small form factor, but why would anyone want to do that?
One reason is to produce low-cost computers. Because these PCs are stripped down to the minimum components needed to have a functioning computer, they tend to be inexpensive. Some, like the Raspberry Pi, don't even have a case or protective covering. The lower prices give people and organizations that normally couldn't afford a computer the option to buy one.
Convenience is another factor. These computers are extremely portable. While they may not have much onboard storage, pairing a mini PC with Web services and cloud storage options can make it a serviceable machine. Gamers won't be rushing out to buy them, and anyone who needs to use resource-hungry software will want to pass them over, but for simple computing tasks they may be the perfect choice.
Some mini PC designers designed their machines with the goal of promoting education. Over time, computers have become more complex, and operating systems are more sophisticated. Operating systems that rely on a graphic user interface (GUI) effectively hide all the processing behind graphics. But with PCs like the Raspberry Pi, all of that complexity is gone.
That means students have an opportunity to learn how programming works from the physical layer on the circuit board to the virtual realm of programming languages. The low cost of the Raspberry Pi and similar computers gives schools and other learning institutions the chance to supply students with a working computer.
The miniaturization trend shows no signs of stopping. In another decade, the phone you carry may put today's fastest home PCs to shame. And who knows? Maybe by then all computers will be small enough to slip into your pocket.