If you really just want a general list of reasons why Rosalina is good (as others have said, she's no where near broken), and still probably one of the top characters in the game even after a patch...
Her aerial hitboxes have huge range, linger for ages, have fantastic priority, and great kill power. Pretty much no one can challenge Rosa's u-air from above her. And it kills early. Dair can spike or alternatively hit off the side of the stage with great power; it's a great gimping move. So is her Fair. Her Nair is crazy. Although 1.0.4. reduced its power, if you use it right, it's very hard for your opponent to punish, because there is little landing lag, and it's active for an insane number of frames. While in midair, you can use it, then watch what your opponent does... you can either push towards them or away from them, you will be able to decide this after her Nair begins, because the attacks lingers for that long. Generally you'll want to follow up on this attack with an immediate d-tilt or f-tilt.
Luma is a 2nd hitbox on the stage. There are fairly obvious advantages to that. And don't fall for Luma's small size, it's deceptively powerful. Luma creates space that can be hard for your opponent to deal with. If they get in Luma's face, Luma can use star bits on them. And if they focus too much on trying to knock Luma way, Rosalina can take advantage of this.
When Luma is attached to Rosalina, her F-Smash has huge range and is very powerful. Her D-Smash has been slightly nerfed, but it covers a lot of ground and it's still very good. Her U-Smash has been nerfed but it's still a reliable KO move.
Rosalina's u-tilt has amazing range. Her D-Tilt covers a lot of ground, you can manipulate Luma to either hit well in front of Rosalina or to hit behind Rosalina. F-tilt is a nice quick attack with good range. She also has an excellent dash attack.
Luma creates a lot of defensive advantages. For one, Luma can literally absorb attacks for Rosalina by just taking the hit. Your opponent can also get stuck hitting Luma, and Rosalina can take advantage of this. Luma can also knock your opponents out of grabs if they hold Rosalina for more than a second, so your opponent simply cannot pummel you if Luma is nearby. Her gravitational pull is also a defensive asset against many characters with projectiles.
The other thing about Luma is that it will help you out at random times. Like, if your opponent getting in a follow up or a combo, Luma will often hit them right out of it. This is an advantage no other character gets, and probably the biggest reason why Rosalina & Luma have been so frustrating for many people to play against.
Her recovery isn't the best because it can be predictable, but it still has good range and is solid overall.
Rosalina does have her weaknesses. For one, she's the 4th lightest character in the game. She's also tall and floaty and this can make it difficult for her when attempting to get back on the ground after she has been hit into the air. She's also much weaker with Luma (although not a pushover) and sometimes it can be very easy to avoid having your Luma knocked off the stage, and each time that happens it's an extra 15 seconds before you'll get Luma back (counting the time it takes Luma to recover).
In summation, Rosalina & Luma's strengths and weaknesses:
Strengths:
-All of her aerial attacks are fantastic
-Great gimping ability
-Her attacks have priority in general
-Great tilt attacks
-Excellent dash attack
-Powerful smash attacks with good range (especially with Luma)
-Long-range recovery
-Luma prevents Rosalina from being pummeled
-Luma will often help Rosalina break free from your opponents combos and follow ups
-Luma absorbs hits
-Luma shenanigans in general (too much to go over here)
Weaknesses:
-4th lightest character in the game
-Luma can easily be knocked off the stage, especially if you're playing near the edge of the stage with your back to the edge.
-Each time Luma gets knocked off the stage, Rosalina will be much weaker for ~15 seconds
-Floatiness and predictable recovery can make it difficult for Rosalina to safely get back to the stage safely after being knocked away
Overall she has more going for her than she has against her, even after the recent patch.
One of the biggest reasons people probably called her "broken" was because before the patch, Luma was able to hit opponents as Rosalina was being thrown, but this has now been removed with the patch. So I expect the cries of Rosalina's brokenness to be subdued going forward.