I don't understand why a class is "public"

I'm beginning to learn C# and I come from a C++ background. The example page I was supposed to create by these instructions looks like

using System.Web;
using System.Web.Mvc;

namespace MvcMovie.Controllers
{
    public class HelloWorldController : Controller
    {
        // 
        // GET: /HelloWorld/ 

        public string Index()
        {
            return "This is my <b>default</b> action...";
        }

        // 
        // GET: /HelloWorld/Welcome/ 

        public string Welcome()
        {
            return "This is the Welcome action method...";
        }
    }
}

My main question is why the HelloWorldController class is prefixed by public. I understand that HelloWorldController is derived from Controller, but why does a class need to be public in the first place? My understanding of the words public and private is that they only have meaning if they're functions inside a class, and that public are the ones that can be used by instances of that class. Also, where is my main.cs in this Visual Studio ASP.NET MVC project that I created?