How do the People that make capture cards make them work like that?
Also Why hasn't Nintendo caught onto this kind of thing and just simply implemented it into their own Consoles???
Others have already gone but I will go anyway.
Three main options
1) A glorified camera attached to the devices in question (probably with a shroud). They actually do this sometimes and if you watch a few video reviews from various places that show the reviewer you might see one.
2) Decode the analogue signal as it goes to the screen and adapt it however you will. It is how
http://home.comcast.net/~olimar/DS/jumbotron/ and the ones that stem from it work.
3) Decode the signal from the actual vram itself and reconstruct it. Going by a few interviews this is what many of the old game magazines used to do to get screenshots and/or remove things to generate maps for areas in the game. This is a bit of a pig to do in real time/proper video capture rates however.
A kind of hybrid of 2 and 3 exists for some of the older home consoles. If they were locked to only output RF or something equally nasty then looking deeper into the console may get you something nicer you can use on a more modern TV with a bit of fiddling.
http://www.gamesx.com/rgbadd/rgbn64.htm is probably one of the more notable examples of this though the original xbox had a VGA mod and even older consoles have similar things.
Later down the line you might also get things like emulators (Nintendo supposedly sent a few to various reviewers at one point) and the handhelds had long had console adapters (the super game boy, various things on the N64 and the gb player on the gamecube) which allowed you to use far more established capture methods.
Why. Because Nintendo devices, much like all devices that are not industrial/field specific, are built to minimise costs and that would add extra.