So, why a dark place and a flash is to be required to freeze the motion (take a high speed photo) though in a bright place with a, say, 1/4000th of a second, we can take a high speed photo?

This is because the flash sync speed, the maximum shutter speed at which the entire image sensor is exposed at the same time, is much slower than 1/4000s. On a typical DSLR, the sync speed is around 1/200s. If you take a photo at that speed in a bright environment, a lot of ambient light will be recorded and you get significant motion blur. If you take the photo in a dark place, there's less ambient light and almost all the light comes from the flash, which is much shorter in duration than the shutter speed.

The reason that the sync speed is so much less than the maximum shutter speed is that most cameras use a curtain shutter which takes time to travel over the sensor. At speeds above the sync speed, the shutter's second curtain starts to cover the sensor before the first curtain has finished exposing it; at very high speeds like 1/4000s, the two curtains are so close that there's only a narrow slit between them to allow light to pass. It still takes around 1/200s for the entire sensor to be exposed, but any portion of the sensor is exposed for only the specified 1/4000s.

If you're taking a photo of something moving quickly, then you really want to expose the whole sensor at the same time. Otherwise, the subject will be moving while different parts of the sensor are exposed, and that causes distorted images. So, you really want to shoot at the flash sync speed, but you want the actual exposure to be faster than that; the way to accomplish that is to remove all the light except the flash.

Why only camcorders that cost too much can record high fps (freeze motion)?

There are really two questions here. I've addressed what I think is your main question, about stopping motion, above. The term fps, though, stands for frames per second. You sometimes see fps used in relation to a given DSLR's ability to shoot in burst mode, where the camera takes photos as fast as it can for as long as you hold the shutter release button. For a DSLR, 12-14 fps is very fast. That rate has little to do with the camera's shutter speed or motion stopping capabilities.

A true high speed camera records video frames so fast that no mechanical shutter could ever keep up. These cameras have single-frame exposure times as short as 1 microsecond and can record hundreds of thousands of frames per second. To make that happen, they naturally need extremely fast sensors and analog to digital converters, multiple processors working in parallel, and hundreds of gigabytes of fast built-in memory. That's a lot of expensive technology aimed at a fairly small market, making it even more expensive.

So, it's important to understand that a high speed video camera is something quite different from a regular old DSLR or camcorder. While you can stop some pretty fast motion for a single image using a flash in a dark room, recording high speed video that would let you see an entire process in extreme motion requires an entirely different approach.