Disclaimer: This trade could not occur as a sign-and-trade over the off-season. In a sign-and-trade, outgoing salary is calculated differently than incoming salary. It would have to occur mid-season once O'Neale has his trade restriction lifted for new contract signings. This is why this trade "fails" on Fanspo.
A summary of this write-up is located at the bottom of the page.
It is well known that the Suns no longer have many trade assets. They can only trade their first round pick on draft night every other year until 2030. Outside of that, they only have 1 second round pick and their current contracts. They are further restricted by the new rules for 2nd apron teams. It will be preventing them from aggregating smaller contracts in a trade, using trade exceptions, using cash considerations, or taking on additional salary in a trade. How will they be able to add talent to improve the team?
They have one option: using bird rights to sign their own free agents to the salary amount needed for a future trade. Bird rights allow teams to retain their own free agents when their current contract expires. If they own a player's full bird rights, they can re-sign them to any amount up to a maximum salary.
The Suns would still need to find an interested trade partner for this player. They can't create incentive through picks, but they can through salary relief. For various reasons, there are teams trying to cut salary in any season. Maybe they don't want to pay a high luxury tax bill, or maybe they want more cap space to pursue upcoming free agents in the following off-season. The Suns can provide this by signing their player to a one-year, expiring contract. Another team can trade a multi-year contract for the expiring contract, and their financial commitment is gone after one season.
The last hurdle is the 2nd apron again. Teams over the second apron are the most likely to be seeking salary relief. But when two teams over the 2nd apron trade with each other, it has to be a 1-for-1 trade where the salaries match precisely. However, thanks to bird rights, the Suns (or any other 2nd apron team) can sign their player to the same annual salary amount as the player another team is attempting to offload.
This trade is an example of this concept in action. Although the Suns and Warriors are both 2nd apron teams, if the Suns signed Royce O'Neale (a free agent this summer, whose bird rights are owned by Phoenix) to Andrew Wiggins' exact salary ($26,276,786), the two teams could trade the two players for each other. The Suns would acquire a much better player, and Golden State would receive salary relief going into the next off-season. The second trade between Boston and Phoenix is an example using this season's salaries. Jaylen Brown and Devin Booker make the same amount this season, so even though Boston and Phoenix are both 2nd apron teams, the trade does not fail for Boston.
TL;DR: Teams can sign their own free agents to any salary up to the maximum if they own that player's full bird rights, so second apron teams can do this to create the contracts they need for salary matching in future trades. Even exact salary amounts in the case of two second apron teams trading with each other. If they sign that player to a 1-year deal, other teams may trade players on longer deals in exchange in order to offload salary.