Blockchains without privacy like Bitcoin only offer pseudo-anonymity. In blockchains without complete privacy, it is the relationships and links between addresses that can reveal private information about you. Every single coin has an immutable history.
This is why freshly mined Bitcoins with no previous transaction history can command premiums of 20% or more as the holder does not have to worry whether it has been tainted and also private.
Zcoin allows you to burn your coins to destroy them so that they stop existing and then redeem them later for coins that have no previous transaction history. The process of burning and redeeming breaks the links between addresses making transaction graph analysis very difficult.
The burning process destroys the coin so that they stop existing and therefore their transaction history stops there and cannot be traced.
The redemption process involves giving a zero-knowledge proof that you previously burnt coins, without having to show which were the coins you burnt. The freshly redeemed coins appear as new coins with no previous transaction history and hence have no linkage with the original coins that were burnt.