It pretty much means the team has 10 days to trade him, release him, or sign him to a minor league contract. If they don't trade him or sign him to minor league contract, he becomes a free agent. If he refuses to sign a minor league contract, he becomes a free agent. If he is traded and the team that trades for him offers him a minor league contract, he can turn down the contract and become a free agent.
A player has all the power in a situation like this. A team designates a player for assignment for two main reasons. The player is out of options, therefore cannot be sent back to the minors without the players consent, and the team wants to place another player on the 25-man and/or 40-man roster. To create the room they need to send someone down to the minors, release a player, or designate a player for assignment. The player has two choices. He can accept the assignment and report to the minor league club or reject the assignment and automatically become a free agent, free to sign with any other club.