I'm sorry, but that video is completely unintelligible; the quality is horrible and I have no idea what the lecturer is saying (I say as someone who's Indian). I couldn't really follow along with his proof, either.
Lets assume that these are all straight marriages. That means that 3 of the 4 women are married to 3 of the 6 men.
Choose one woman out of 4, and choose one man out of 6. They are now married and accounted for 4x6 = 24 possibilities. Then choose another woman out of the remaining 3 women. She is married to one of the 5 remaining men. There are 3 x 5 = 15 ways to do this, so we now have a total of 15 x 24 = 360 choices. Then choose another woman from from the remaining 2 women, and assign her one of the 4 remaining men. That yields 2 x 4 = 8 possibilities, so there are 8 x 360 = 2880 possible couples. This matches with what you calculated I believe (though Reddit seemed to have formatted what you said incorrectly).
Edit: In fact, you need to divide by 6 to account for the ordering possibilities, as you could have chosen the 3 couples in different orders, and we need to avoid double counting. There are 6 ways to order 3 things (3 choices of the first thing, 2 choices of the second thing, and 1 choice for the last thing to determine the ordering, so 3x2x1 = 6 choices of ordering). Thus, the correct answer is 2880/6 = 480, which is 6x5x4x4. I still maintain that the proof given in the video was impossible to follow, though it does yield the correct answer.