As I understand it, it means that a valid transaction is accompanied by a zero-knowledge proof, proving that this transaction is in fact valid, but revealing nothing about source or destination of funds.

Bitcoin requires each user to verify the entire blockchain, whereas the SNARK-technique means a single user can verify the blockchain, and produce a piece of data which essentially proves that 1. the verification has taken place and that 2. the claimed result of the verification (valid/non-valid) is the actual result of running the verification.

If it sounds too good to be true, that's because it is fairly revolutionary (if it works). Here's the paper: https://eprint.iacr.org/2013/507.pdf