Unity 5.2 default 3rd person controller makes character fly upwards when moving forward

Question

I'm new to Unity, and already dealing with a problem using the default third person controller script. I am trying to move and animate the player, but I have a problem, as when I try to move the character it moves beautifully but not only vertically and horizontally - it also moves up on the y axis! It just keeps going higher and higher, and though it's quite funny, it really messes the gameplay.

Here are screenshots of what happens, and yes, I do have gravity turned on:

Answer 1

Since they original code is not posted on this site, I am not going to click on the link. Instead, I will give you a very simple 3rd person controller script that illustrates how movement can be handled on a basic level. And to be honest, animations are much better handled using the Animation Controller Component. It uses a state machine to control the animations and you can add int parameters. In addition to the simple movement in 3rd person view, I will also comment on how you could take advantage of the Animation Controller in scripting.

using UnityEngine;
using System.Collections;

public class ThirdPersonCamera : MonoBehaviour
{
    public Camera cam;
    public GameObject character;
    public Animator anim; // Gets the Animator component and
    // is better used when it is attached to the character

    void followCharacter()
    {
        if (Input.GetKey(KeyCode.LeftArrow))
        {
            cam.transform.position -= new Vector3(1, 0, 0);
            //anim.SetInt(1); if the parameter is an integer
            // it will move to that specific animation
        }

        else if (Input.GetKey(KeyCode.RightArrow))
        {
            cam.transform.position += new Vector3(1, 0, 0);
        }

        else if (Input.GetKey(KeyCode.UpArrow))
        {
            cam.transform.position += new Vector3(0, 0, 1);
        }

        else if (Input.GetKey(KeyCode.DownArrow))
        {
            cam.transform.position -= new Vector3(0, 0, 1);
        }
    }

    void Update()
    {
        followCharacter();
    }
}