I am having a hard time seeing just what makes the animator change states.

I have my animations ready, however to make them change when my character walks, runs etc. is something I don’t get.

I’ve watched the video, but it doesn’t seem to cover all of it. Just the basics.

This tutorial should help you get the basics of what you need to do:

Here's the script
  Animator anim;
  // Use this for initialization
  void Start () {
      anim = GetComponent<Animator>();
  // Update is called once per frame
  void Update () {

      float input_x = Input.GetAxisRaw("Horizontal");
      float input_y = Input.GetAxisRaw("Vertical");
      bool isWalking = (Mathf.Abs(input_x) + Mathf.Abs(input_y)) > 0;
      anim.SetBool("isWalking", isWalking);
          anim.SetFloat("x", input_x);
          anim.SetFloat("y", input_y);
          transform.position += new Vector3(input_x, input_y, 0).normalized * Time.deltaTime;