You don't use them, you just collect them.
Certain badges grant your Pokémon certain things such as more attack, defense, speed, etc.
Also, if you get more badges, you will be able to use HMs (Hidden Machines such as Cut, Fly, etc.)
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No, you can trade with no badges at all.
The badges are just,there for you to be able to use HMS outside of battle and for high levels of Pokemon to listen to you depending on the badge...
who's blue. Whatever here's how you have to get 7 gym badges in kanto then he'll battle you.
in total 16 badges
in order for trader Pokemon to listen to you. you need gym badges if you have al 8 badges they will obey and if you have 3 or four badges is going to e hard to train Pokemon