stars and what do they do?

[Lone Wolf]

do they improve anything?

[WonderSparks]

Yes, each star grants a boost to the Pal's stats, and at four stars their work suitabilities are also improved by one level. (Only their existing suitabilities, tho; a Foxparks doesn't magically gain the ability to water your plants, for example)

[SidBangBang]

stars also greatly improve the pal's partner ability (the ability that unlocks when you make saddles, gloves, belts, etc for that pal species).
for pals that ranch like the chikipi or vixy, it increases the number of items they drop.

[saiyuk976]

What's already mention, with emphasis on the partner skills.
Verdash for movespeed bonus, higher bonus at higher star levels
Vanwyrm damage bonus to weak points while mounted
Ragnahawk adds fire damage to and increase damage of attacks while mounted
I think vanwyrm goes from 20% to 40% but ragnahawk is 50% to 100% bonus damage, which along side a damage trinket and some points in attack and damage buff food you can put out some nasty damage with a decent weapon.