The oldest reference I could find is from August 2018, when Gauntlet RPG proposed Stars and Wishes as a softer alternative to the approach of "Roses & Thorns", which also is at times worded as "Rose, Thorn & Bud".
Roses & Thorns is meant to identify "I liked this, I disliked that" and possibly "This has potential".
a variant of Rose Thorn & Bud in action
Oddly enough, even before Stars and Wishes became a thing, I was introduced to a similar system around 2010 in a table-round of WoD. The GM called it maneuver critique or at other times "Wishes, Notes, Critique". It's a variant of Rose, Thorn & Bud method.
Starting with the player right of the GM, each player would nominate one or more actions of other players that they found memorable, remarkable, or cool, and thought would be worth experience points for playing the character or showing character growth. You could not nominate your own character this way. After a nomination, one other person at the table (GM included) had to agree, turning the nomination into an XP. Only after the other players, the GM would dole out first other nominations for the players to vote on, and then points he definitely gave to players. Generally, most characters or players would get one or two nominations, and it was quite rare to leave the table with just the experience points for survived the session.
While this maneuver critique had a positive reinforcement for good actions (stars), it was at times used for some wishes but did neither require it nor give those any specific spotlight. In contrast to Stars and wishes, the Maneuver critique could and would at times be formulated rather harshly and in some cases accusatory in the critique part. Which indeed can hurt - to the point that a long-standing player stopped attending after people critiqued repeatedly that they are not comfortable with something.
Meaning of Stars & Wishes
Stars are simple: Every involved person shall point to one item, person, or incident that they really enjoyed during the session. It could be "I really loved how you portrayed that Rat-Chaos-Mutant's speech, GM-Bob." or "Christy, your Barbarian was so cool when she burst through the door yelling bloody murder." Just name something that you really enjoyed.
Wishes are not so simple: It's formulating about what you like to see in the future. But it should be constructive.
For example "Please, no more zombies" is not a valid wish. You need to formulate your wish in a positive light on how to make things better. The wish for different enemies should be formulated in a way to what you want to face, not just what you don't want to face. "With all those zombies around, can we finally face the necromancer making them?" or "I think fighting a larger boss monster instead of just T-virus Zombies might be a nice change."
Wishes also allow asking changes that players can do or that were harming your fun in the game in a positive manner. "Christy, could you try to not yell your battle cry into my ear next time you do your battle cry?" At times it can be used to suggest things about things that affect the game but are not in the game entirely. "I'd like if you'd drop a note if you are late in the future" is a valid wish in this regard.
While please get better and stay safe is a very nice wish to have, it isn't entirely constructive. Still, one game recently had made almost all of us wish for health as more than half of us had to cut short (compared to other rounds) due to health reasons. Unless you really have nothing else to wish for, you shouldn't use that one wish you have just for health.
In my experience, it helped to break the ice with the system if the player that introduced the system goes ahead and starts with it. When the system was introduced to me, the one bringing it up went ahead and started with making their stars and wishes that round. That was some year and a half ago. It also did help that I knew a similar system before.
One of my mainstay rounds is play-by-chat. In those sessions, we present our stars and wishes simultaneously, not in rounds as Gauntlet proposes. We put up a dedicated channel for those, and besides other almost ritualistic things at the end of the game, the words "don't forget the stars and wishes" in the general channel have become a part of the closure rites in this round.
In a recent, short running group via voice chat, I just introduced the system at the end of the session with a very quick rundown of what is the system meant to do, and surprisingly, the GM jumped the wagon and said "Know what? good idea! Why don't you start?" After setting the example, the others followed to name their most liked moments and what they'd like to have a little focus on next time. In a similar fashion to the play-by-chat example, we only had one call to everybody to name the Stars and Wishes together to keep us a little shorter - we had deadlines to keep.
in a more direct session, asking some questions at the outset like "Did everybody have fun?" and "Are you ok with the pace?" by the GM can help to open up, allowing players to get into the mindset to answer constructively before pointing out their stars.