Insert/Update Many to Many Entity Framework . How do I do it?

Try this one for Updating:

[HttpPost]
public ActionResult Edit(Models.MathClass mathClassModel)
{
    //get current entry from db (db is context)
    var item = db.Entry<Models.MathClass>(mathClassModel);

    //change item state to modified
    item.State = System.Data.Entity.EntityState.Modified;

    //load existing items for ManyToMany collection
    item.Collection(i => i.Students).Load();

    //clear Student items          
    mathClassModel.Students.Clear();

    //add Toner items
    foreach (var studentId in mathClassModel.SelectedStudents)
    {
        var student = db.Student.Find(int.Parse(studentId));
        mathClassModel.Students.Add(student);
    }                

    if (ModelState.IsValid)
    {
       db.SaveChanges();
       return RedirectToAction("Index");
    }

    return View(mathClassModel);
}

In terms of entities (or objects) you have a Class object which has a collection of Students and a Student object that has a collection of Classes. Since your StudentClass table only contains the Ids and no extra information, EF does not generate an entity for the joining table. That is the correct behaviour and that's what you expect.

Now, when doing inserts or updates, try to think in terms of objects. E.g. if you want to insert a class with two students, create the Class object, the Student objects, add the students to the class Students collection add the Class object to the context and call SaveChanges:

using (var context = new YourContext())
{
    var mathClass = new Class { Name = "Math" };
    mathClass.Students.Add(new Student { Name = "Alice" });
    mathClass.Students.Add(new Student { Name = "Bob" });

    context.AddToClasses(mathClass);
    context.SaveChanges();
}

This will create an entry in the Class table, two entries in the Student table and two entries in the StudentClass table linking them together.

You basically do the same for updates. Just fetch the data, modify the graph by adding and removing objects from collections, call SaveChanges. Check this similar question for details.

Edit:

According to your comment, you need to insert a new Class and add two existing Students to it:

using (var context = new YourContext())
{
    var mathClass= new Class { Name = "Math" };
    Student student1 = context.Students.FirstOrDefault(s => s.Name == "Alice");
    Student student2 = context.Students.FirstOrDefault(s => s.Name == "Bob");
    mathClass.Students.Add(student1);
    mathClass.Students.Add(student2);

    context.AddToClasses(mathClass);
    context.SaveChanges();
}

Since both students are already in the database, they won't be inserted, but since they are now in the Students collection of the Class, two entries will be inserted into the StudentClass table.