Zero knowledge proofs themselves do not provide privacy, encryption does. The details of a transaction are encrypted so an outside observer cannot know the sender, receiver, amount, or the message contents. Privacy is further provided in that every transaction, looks identical.
The problem is, in a public blockchain the nodes must be able to verify the transaction. For example, the amount of the input must match the amount of the output (otherwise you are creating or destroying "coins"). On a transparent chain this is easy because the nodes can see for themselves. However, if the transaction is encrypted, how could the network know a transaction is proper?
That is where the zero knowledge proofs come in. They prove to a verifier (other nodes), that the contents of the encrypted transaction meet the network rules, without revealing the contents of the transaction itself. For example, the zkp is proof the amount of the input exactly matches the amount of the output, without having to know the amounts. As a verifier, I can know the amount of your input, matches your output exactly, without having to know any other details.
Privacy is in your behaviors. Zero-knowledge proofs make it possible for an encrypted transaction to be verified by the network on a public ledger.
