A team is allowed to waive one player without having the remaining portion of the contract count towards the salary cap and/or luxury tax.
Teams can bid on a player that has been amnestied. They are allowed to bid by only what space they have under the cap. The highest bid claims the player and the team that placed the player on waivers via amnesty is required to pay the remaining portion of the salary (not counted against the cap).
Example: Player A is waived from Team A via the amnesty clause and has $12 million owed. If Team B bids $2 million and Team C bids $4 million, Team C would claim Player A off waivers for the $4 million. Team A would be required to pay $8 million to Player A, but does not count against the cap.