From the Super Mario Wiki
The Red Star is an item that appears only in Super Mario Galaxy. It resembles a red Grand Star and holds the power of the Red Lumas, one of whom first gives Mario the Red Star in the Gateway Galaxy. It transforms Mario into Flying Mario, who has the ability to fly through the air, similar to Wing Mario from Super Mario 64. However, unlike the effects of the Wing Cap, the Red Star allows Mario to fly more freely, changes the color of his clothes, gives the player no wings, and is self-propelled, like the P-Wing, and also would grant Mario a gravitational pull when he spins pulling in star-bits, and coins on ground or in flight.
After obtaining the Red Star, Mario's clothing changes. Instead of his normal red cap and blue overalls, he wears a black cap and black overalls, while his red shirt remains unchanged. While flying, Mario's hands glow, and he leaves red streaks in his path. By holding , Mario will stop in midair, and red streaks will point in the direction he is facing. After collecting a Red Star, flight is activated by executing a Star Spin in the air, as opposed to performing a Triple Jump, as is required with Wing Mario.
Red Stars play a very limited role in the game. Other than allowing Mario to complete a Purple Coin challenge in the Gateway Galaxy, the only other appearance of any Red Stars in the entire game is in the Comet Observatory, where they can be used for exploration, and for reaching a few 1-Up Mushrooms that are otherwise inaccessible, such as the one on top of the Garden spire and the one beneath the Gate.