The player first is removed from your ML team and the 40-man roster and put onto waivers. At which point, neither you nor the owner of the franchise you drafted the player from can put a claim in on him.
If he is claimed by another owner, then he and his special Rule 5 status are transferred to the new club. If he isn't claimed, he will be sent back to his original franchise.
They then have 2 days to either accept the offer back for $25,000 or decline it. If they accept the offer back, the team that drafted him loses all rights to the player. If the original franchise decides to decline the option, the player then loses his unique Rule 5 status and becomes a player with no additional rights. Either way, if the player gets sent back to his original franchise for them to decide to accept or decline the option, the player will lose his Rule 5 status.
If a Rule 5 pick is acquired via trade or waiver claim, the new club has 72 hours to assign the player to his big league team or else the whole Rule 5 Offer back process starts again.