Brains decide what a Character wants to do so that other components can execute those actions. For example, a brain would choose which direction to move and when to jump, then the Character Movement component moves them in that direction and the Jump component launches them into the air and plays the appropriate animations. Animancer's Brains and More Brains examples explain the concept in more detail.

Each type of brain Inherits from the base CharacterBrain to make use of its DefaultExecutionOrder (so it can determine how it wants to control things before everything else executes those actions) and a reference to the Character, but there is nothing special about it so any script could potentially control a Character:
public abstract class CharacterBrain : MonoBehaviour
    public const int DefaultExecutionOrder = -10000;

    private Character _Character;

    public ref Character Character => ref _Character;

    protected virtual void OnValidate()
        gameObject.GetComponentInParentOrChildren(ref _Character);

Player Brain Controls the character using keyboard and mouse input.
Behaviour Tree Brain Controls the character using a very simple Behaviour Tree system to allow enemy behaviour to be built from small modules instead of hard-coding a new script for each one.