@DrewDad,
OK, I watched the video.
"Begouch" or "begoush" (maybe a take on "by gosh"?) is a made-up word that they use in the video. It's used like an exclamation or curse word (damn, hell, etc.), but they used a made-up word so that they didn't have to censor the video.
The person in the comments wrote lines from the video that he or she found particularly funny.