alt text

Code:

using UnityEngine;
using UnityEngine.UI;
using System.Collections;
using System.Collections.Generic;

public class Manager : MonoBehaviour 
{
    bool isChatting;
    bool sendText;

    public Text chatText;

    List<string> chatMsg;

    private string display = "";

    string playerName;

    public GameObject menuPanel;
    public GameObject chatPanel;
    public GameObject optionPanel;
    public GameObject multiplayerPanel;
    public GameObject joinPanel;
    public GameObject createServerPanel;
    public GameObject pausePanel;

    void Awake()
    {
        chatPanel.SetActive(false);
    }

    void Start()
    {
        playerName = "SuperHB";

        chatMsg = new List<string>();
    }

    void Update()
    {
        if(Input.GetKeyDown(KeyCode.Y) && isChatting == false)
        {
            isChatting = true;

            chatPanel.SetActive(true);

            sendText = true;
        }

        if(Input.GetKeyDown(KeyCode.Escape) && isChatting == true)
        {
            isChatting = false;

            chatPanel.SetActive(false);

            sendText = false;
        }

        if(Input.GetKeyDown(KeyCode.Return) && sendText == true)
        {
            OnSend();
        }
    }

    public void Msg(string writtenMsg)
    {
        chatMsg.Add(writtenMsg.ToString());
    }

    public void OnSend()
    {
        foreach(string msg in chatMsg)
        {
            display = display.ToString() + playerName.ToString() + ": " + msg.ToString() + "

";
}
chatText.text = display;
}
}

Hierarchy Close-up:

alt text

This is happening as you are using an array to store the newly sent messages, and adding the list to the chat box each time. Instead you should store it in a string, and just add this new string to the chatText.text. After this you can add the string to the array.

Example:

chatText.text += "

" + newMessageThatWasSent;
//Then if we want a record of the messages in an array we add this new message to it.
chatMessagesLog.Add(newMessageThatWasSent);

Also you don’t need to use ToString() on a string.