Local and global coordinate system

I've started to learn Unity and I'm not sure how local and global coordinates work. I have two scripts.

First (responsible for character movement (A,W,S,D)):

public class Move : MonoBehaviour
{ 
    // Start is called before the first frame update public float speed = 6.0f; private CharacterController _charController; public float gravity = -9.8f;
    // Update is called once per frame

    void Start()
    {
        _charController = GetComponent<CharacterController>();
    }

    void Update()
    {
        float deltaX = Input.GetAxis("Horizontal") * speed;
        float deltaZ = Input.GetAxis("Vertical") * speed;
        Vector3 movement = new Vector3(deltaX, 0, deltaZ);
        movement = Vector3.ClampMagnitude(movement, speed);
        movement.y = gravity;
        movement *= Time.deltaTime;
        //movement = transform.TransformDirection(movement);
        _charController.Move(movement);
    } 
}

And second (mouse rotation):

public class MouseLook : MonoBehaviour
{ 
    public enum RotationAxes {MouseXAndY = 0, MouseX = 1, MouseY = 2 }    

    public RotationAxes axes = RotationAxes.MouseXAndY; 

    public float sensitivityHor = 9.0f; 
    public float sensitivityVert = 9.0f; 

    private float _rotationX = 0; 

    public float minimumVert = -900f; 
    public float maximumVert = 900f;

    void Start()
    {
        Rigidbody body = GetComponent<Rigidbody>();

        if (body != null)  
            body.freezeRotation = true;
    }

    void Update()
    {
        if (axes == RotationAxes.MouseX)
        {
            transform.Rotate(0, Input.GetAxis("Mouse X") * sensitivityHor, 0);
        }
        else if (axes == RotationAxes.MouseY)
        {
            _rotationX -= Input.GetAxis("Mouse Y") * sensitivityVert;
            _rotationX = Mathf.Clamp(_rotationX, minimumVert, maximumVert);
            float rotationY = transform.localEulerAngles.y;
            transform.localEulerAngles = new Vector3(_rotationX, rotationY, 0);
        }
        else
        {
            _rotationX  -= Input.GetAxis("Mouse Y") * sensitivityVert;
            _rotationX = Mathf.Clamp(_rotationX, minimumVert, maximumVert);
            float delta = Input.GetAxis("Mouse X") * sensitivityHor;  
            float rotationY = transform.localEulerAngles.y + delta;  
            transform.localEulerAngles = new Vector3(_rotationX, rotationY, 0);
        }
    }
}

It is not clear to me exactly how the local and global coordinate systems work and interact here. What exactly happens when we translate from local to global coordinate system

movement = transform.TransformDirection(movement);

How my mouse rotation works here is also not entirely clear. As I see it, it's in local coordinate system, so how does the code work so that when you turn the mouse, you will move straight ahead depending on where you turn it?

In summary, I don't understand how translation from the local to global coordinate system works here, how these coordinate systems work by themselves, and why translate from one to the other at all? And how dependency works: which way you turn the mouse, that's where you go straight.

The book I'm reading now doesn't seem to have explained it very well, so I'm hoping for some help.

Local calculations are relative to the object they're being calculated for, while global (or world) calcs are always fixed to the world. For example, lets say you were a character in your game; you're on a beach and facing east, towards the sea. Your own Z forward would face east, but the world's Z forward always faces north, regardless of what direction you're facing, like a compass. So: world = fixed constant, local = relative to you. You use these as required - for example if you were scripting a car to go forward you'd probably use local transform.forward because you want to go in whatever the direction the car is facing. If you wanted something to point north you'd probably use world Vector3(0, 0, 1) because it's a fixed point.

It's hard to tell from your script which you need but you should almost certainly avoid assigning rotations directly e.g. localEulerAngles = in favour of the built in methods e.g. transform.rotation = (https://docs.unity3d.com/ScriptReference/Transform.html).