Answer 1

The videos you see on YouTube are taken from Emulators playing Pokemon ROMs.
Here's one video I found that tells you what software you can use if you wanna do it too: http://www.youtube.com/watch?v=33Qd-klIxlY

Otherwise you'd need to do what Speed freak said and set up a camera to record off the DS. You need to do that for 3DS at the moment since there is no emulator for it yet.

Answer 2

You would have to set up your Camera in a certain position in which it can see the DS system's Screen precisely. That is really the only way, but make sure you have good lighting in order to prevent blurriness. Something to know is that from Gen IV onwards there is a Vs. Recorder that can record a Battle Frontier/Subway or Wi-fi battle and keep it there, but you can only have one at a time. Then you film that battle without having to press button which might interfere with the Camera, it also speeds up reaction time.

Comments

No I mean good quality vids like this http://www.youtube.com/watch?v=EUWSPpOlx2Y&feature=plcp&context=C3d6ac87UDOEgsToPDskIFzblEHQ9ZmVKiouzJFPE1

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Hex, most of the time (probably 90% of the time(, the ones with good quality are recorded with emulators on a computer, because it is a lot easier to record on.

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but how to they get the battles there? They do play the battles on DS.