You can purchase a video capture card, newegg.com and tigerdirect.com usually have good prices on them. The trouble is, you have to hook the system up to your computer in order to record them on your pc. There isn't a way to 'transfer' it over network or XBox Live, nothing like that.
As for the user who said you get the PC version.. There isn't one. And as far as official notice goes, there isn't even WORK being done on porting it over. Halo 1 and 2 didn't sell well on the PC, so Microsoft may have decided to keep it strictly to the 360 for now. That's not to say they won't change their mind in the future, but Bungie has their hands full, and gearbox(The company that ported the last two Halo games to pc) has other projects that they're working on too. The odds of a pc version don't look good.
So, you buy the capture card, hook it up, then save your replay on your 360, plug it in to your pc, hit record on your computer, and play back the recorded replay from your system.
Hope that helps!