I read that a Phantom will record footage at about 2570 fps @ 1080p. And a high end bridge camera would capture stills at 1/4000th of a second. 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?... why one need an expensive high speed camcorder to do it otherwise?
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.
Because there's a lot more to "taking a photo" than just opening and closing the shutter. The data has to be read off the sensor, processed to a greater (JPEG) or lesser (RAW) extent and a greater (RAW) or lesser (JPEG) amount of data written to storage. Doing that once, then waiting a second and doing it again is relatively easy - and that's what bridge cameras do; maybe 2 or 3 fps. Doing it every 1/2500s, consistently and on time, is hard and that's what you pay the big bucks for.
The refresh rate aka readout speed (basically how quickly the sensor spits out all the data from the current frame and is ready to record the next) of most sensors is surprisingly slow. So slow that many require a mechanical shutter to control short exposure times.
In other words: to take a still photo at 1/4000s, you simply close that mechanical shutter after 1/4000s to prevent any more light from being recorded and nevermind that the sensor needs several moments longer than that to finish outputting the data and get ready to fire another shot.
A motion camera is different, of course. In that case, the sensor actually has to actually finish recording the first frame and be ready to record the second. As has already been mentioned, one reason that's so much more expensive is that - particularly for a high speed camera - that requires a lot more memory and processing power "behind" the sensor.
But that's actually not the primary factor. Design and construction of a sensor is extremely expensive. Still cameras are affordable only due to economy of scale (note that many or most cameras share just a few sensors from just a few manufacturers - practically just Sony at the high end). But as we've just described, motion cameras require different sensors. And motion - particularly high speed, again - is a far smaller market which can't achieve the same economies of scale. So a motion camera must be very expensive or, like VDSLRs, use a regular "stills" sensor and suffer from
rolling shutter as a result of slow refresh rate.