If you're stubborn, it's hard for other people to help you or change your mind. Stubborn people keep on doing the same things, or keep on believing the same things, and they won't listen when other people try to help them, or try to change what they're doing to make things better for them.
This dog is being stubborn. He does NOT want to walk, and you can't make him! He's just sitting there stubbornly. He won't listen, and he won't change his mind, even if his human says, "Come on, let's walk together! It'll be fun! It'll be good exercise!" Nope. This dog is stubborn.
If a mess or a stain is stubborn, you've tried and tried to clean it, but it just stays and stays. And if a problem is stubborn, you've tried and tried to solve it, but the problem just continues and continues, or even gets worse. What if your cough is stubborn? Yup: it means you cough and cough, day after day.
In Green Eggs and Ham, both of the characters are stubborn! Sam stubbornly pushes his friend to try the green eggs and ham: "Try them! Try them! Try them!" And his friend stubbornly refuses to try them. "No, no, no! I do not like them!" If you've made up your mind and you will NOT change it, or you want something to happen and you will NOT let it go, even when people have told you "no," you're being stubborn.