If you give a Qualifying Offer to a Free Agent, it's like offering him a one year $15.8 million offer. The player can accept or reject. As of this year, no player has ever accepted the one year offer. If a player rejects, then he is free to negotiate like any other free agent with other teams. The big catch with these players is, if a team signs them, they forfeit their lowest unprotected pick. I believe the protected picks are 1-10. Might be 1-15. The team that offered the QO (Former team) will get a sandwich pick between the first and second round.
So in the Angels case, we did not offer a QO to Freese. So if he signs with another team, we do not get the sandwich pick. But if we sign a player that received a qualifying offer, even if it's for say 3 years/$30 million, we would lose our lowest round pick. And if we sign multiple, we would lose our next pick. I think there is a limit we can lose, and anything over would carry over to next season. So if we go crazy and sign like 5, we would lose so many this year, and so many next year.