Where can I see ProPhoto RGB photos? Do I need a special monitor? Can I print it? If I print it how would it look (will it look like in the special monitor)?
You can see them anywhere really. To get the most accurate view, they have to be converted to the colorspace of the output device, otherwise you will see a very flat and dull looking image. ProPhoto RGB images do not contain all colors from that color-space but are allowed to contain colors that cannot be represented by other more narrow ones.
The point of ProPhoto RGB is to offer a huge color-space so to minimize loss of color that was captured in an image. Those images will not use all colors in ProPhoto RGB but colors within the gamut of the input device. By converting to ProPhoto RGB, images should retain most of their color-gamut and therefore be more accurate when converted to the color-space of the output device.
The greater the difference between the input and output color-gamut, the more advantageous it is to use such a wide intermediate color-space. Printers have very different color-gamuts than displays and with now 10+ colors to mix on high-end printers, they can represent colors that would get lost if converted to something like sRGB.
The ProPhoto colorspace cannot be displayed in its entirety. According to Wikipedia:
One of the downsides to this color space is that approximately 13% of the representable colors are imaginary colors that do not exist and are not visible colors.
ProPhoto is useful as a working colorspace when the full gamut contained within images needs to be preserved. However, if care is not taken while editing, banding and conversion artifacts may be introduced.
Images that are not exported to a conventional colorspace for display will have a desaturated appearance when viewed in environments that are not color managed. In environments that are color managed, they should look "normal".
The ProPhoto color space is just another way of describing RGB colors. sRGB, AdobeRGB, and ProPhoto can all describe/display the same colors (where they overlap/using different numbers); but the larger color spaces can describe/display colors the smaller spaces cannot without using negative values. But just because a color space can describe a given color does not mean an image/scene contains that color.
A monitor is an RGB device, and if you have a monitor that calibrates as "100% AdobeRGB" that does not make it an AdobeRGB device... it has its' own profile. The output from a digital camera is RGB as well, but its' input does not have a color space at all.
So take a digital camera that can record (react to) all visible light, and even some light that is invisible (IR/UV), in raw... that raw file output would exceed all color spaces in some aspect. You then edit that file in Lightroom (ProPhoto color space) and display it on your 100% AdobeRGB monitor. The monitor space/gamut/capability is the limit of what you can see; but it is much greater than sRGB.
Or you use the camera to record a jpeg, and you set your camera to use the sRGB color space instead of the AdobeRGB. So that same image is now limited to sRGB no matter where/how it is displayed. And if your monitor calibrates as 98% sRGB you would probably never see a difference. But if printed it would not take advantage of the full CMYK color space/capability; whereas an image that was originally in the ProPhoto color space could (assuming the original image contained those colors). Also note that a monitor can calibrate as "98% sRGB" yet still be able to display colors outside of sRGB... e.g. your monitor's gamut could look something like the CMYK gamut (it almost certainly does not look like the sRGB triangle).
The point of recording as much as possible (raw files), and maintaining as much as possible (ProPhoto), is to take advantage of those gains where possible now; and to enable potential further advantage in the future. Currently there are no monitors that can display all of the visible colors w/in ProPhoto, but there may be someday. But there are currently many monitors and printers that can exceed the capability of sRGB (however, to take full advantage of that requires effort).