You don't have a 6700 cassette - the lockring is clearly labelled: 7800, meaning Dura-Ace, unless someone inexplicably swapped out your lockring. The cassette doesn't look particularly worn, although if the original owner of the bike was putting in any decent miles, the chain should have been replaced at least a dozen times since purchase, and the cassette replaced 4+ times.
7800 cassettes new cost around $250 when you could get them; even used now they will fetch $50 at least. I am sure your local shop would be pleased to take this used and worthless cassette off of your hands for replacement. Then out the back door onto Ebay for mechanics beer fund money.
Are all of the cogs skipping, or only a few? Or is the problem with the chainrings? Or a mangled or mis-installed quick link? Too many vague clues here.