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Yes I'm pretty sure it's 6!. You lock in the host and hostess, and then arrange the guests around them. The guests can be arranged in 6! ways. The reason why we don't multiply 6! by 2! (due to the misconception that the host and hostess swapping places will give new cases) is because of this...

If you're talking about the highlighted part: For LHS, you use the binomial theorem formula, and on RHS, you expand nCk into factorial form and multiply it by 1/n^k. The reason why it ends at (n-k+1) is because the remaining terms cancel out with (n-k)! in the denominator