Video cameras can work at much higher ISO than still cameras for two reasons:
1. Video cameras work with much lower resolution than still cameras. Even the best HD video cameras only record at 1920x1080 resolution (just over 2 megapixels), and standard definition cameras are generally 720x480 (just over 1/3 of a megapixel). If one were to take an unacceptably noisy shot from a DSLR and size it down to these resolutions, it would look reasonably good.
2. In video, each frame is only displayed for a short time, 1/30 of a second or so, before being replaced by the next frame. The noise in successive similar frames tends to average out over time, and is perceptually more difficult to see. You can see this for yourself by playing a DVD on your computer, and periodically pausing playback and looking for image artifacts. You'll find things when the image is paused that you simply won't see during playback.