Rewind back to the first "Super Mario Bros." game, three powerups existed in the game. Nothing compared to the several featured in Super Mario Bros. 3. In number 1, the featured powerups was the mushroom, fireflower and the starman. By far, the star was the most powerful. It would allow Mario or Luigi to kill the enemy by touch. But one must wonder... does the star man grant invincibility? Mario could survive anything... except holes, lava and sometimes water. Now Mario was invincible. Except to those. Maybe there IS a reason.
In Super Mario Bros. 2 and 3 the star worked the same way, as well as Super Mario World. In Super Mario RPG, it was coveted as a quick way to gain experience. But how can simply touching the enemy be enough to give experience, with only simply brushing by them? In the Mario Kart series, they caused the other drivers to spin out.
Instead of droning on and making a short thing long, I'll make it blunt. The star increases speed and strength. While it does not allow him to run super fast, it allows him to blast away a group of enemies in record time (especially Super Mario RPG). It does, however increase the speed of a go-kart. But so do mushrooms (both types), which could lead to special properties of the karts (which need not be covered now). In Yoshi's Island, baby Mario became even stronger than yoshi - he ran at amazing speeds and could destroy any enemy.
As stated before, it will send Yoshi to growth instantly. But, it was not mentioned that feathers, fire flowers, mushrooms and 1-up mushroom do the same thing. If it can be explained, increasing Yoshi's strength and speed would allow the already quickly growing creature to reach maturity even faster.
Any relation to the Star Spirits is unknown. They are somewhat similar in appearance to starkids, but have never directly spoke. If anything, these would be the lesser of the star species.