Here's the simplest explanation for renouncing rights:
Originally Posted by Addict4Sneakers
What does renouncing arenas in 2015 mean?
There are all sorts of cap holds but the ones you're talking about are sometimes called "free agent cap holds." Any player that a team has rights to, like "Bird Rights" or "Early Bird" rights, is assigned a value based on their last contract. That's the "hold" you see. It is only counted against the salary cap, not the luxury tax. Rights don't go away on their own so when a player retires (or can't find a new team) the last team that had them under contract can theoretically hold those rights until they die.
A team can renounce their rights to the player and the hold goes away. You have to file paperwork to do this. If your active contracts are already over the cap there's no reason to do this so teams just don't file the paperwork and the holds don't go away. There is a tiny value in having them that in certain circumstances it's actually better to stay over the cap instead of temporarily going under so teams keep them just in case, and because no one likes doing paperwork they don't have to.
If the Celtics go under the cap this offseason and want to sign a free agent they'll just file the paperwork to renounce the holds and they'll all disappear that day.
Basically, renouncing a player gets rid of a cap hold. Teams only renounce rights when they're under the cap so they can free up space to sign players. The Celtics recently renounced a bunch of random players from a long time ago, like Shaq, Marbury, Dana Barros, Scott Pollard, Michael Olowokandi (yes lol), etc. They were a luxury tax team for years (with the Big 3 and others), so it didn't matter, as if they ever wanted to re-sign one of those players they could go over the cap (thanks to Bird Rights). This year, they wanted to sign Amir Johnson and Jonas Jerebko, and they were under the salary cap, so they renounced a bunch of players to get rid of those cap holds.