dictionary output problem

First of all, and probably the most important thing to understand: a return statement not just returns a specified value (or None), but exits the function immediately. You can have multiple possible "exit points" (return statements) in your function, but you can only exit once. When you reach a return statement in your function, it means it is the end of that function.

for key in tchrs.values() --> The naming is confusing here. Since you’re iterating through the values of the dictionary, then it’s logical to name that temporary variable value. In our case, the dictionary values, which we are iterating through, are lists of courses (of one teacher), so classes, class_list, or something like that might be even better (but key should by no means be used).

We are interested in the number of courses, i.e. the length of those lists:

        ...
        for classes in teachers.values():
            if len(classes) > max_count:
                max_count = len(classes)

Now we have max_count, but we’re not finished: we are looking for the name of the teacher, i.e. the corresponding dictionary key. (Remeber that dict[value] will give back the corresponding key if you have the value.) Note that we cannot return the name immediately inside the loop, since in that case we would exit the whole function right after the first iteration. So we have to save it in a variable, but this variable (just like max_count) has to be initialized outside the loop:

def most_classes(teachers):
    max_count = 0
    name_of_teacher = ""

    for classes in teachers.values()
        if len(classes) > max_count:
            max_count = len(classes)        
            # teachers[classes] will give back the corresponding key i.e. the name      
            name_of_teacher = teachers[classes]  

    return name_of_teacher

This one is not shorter or simpler, but we can also use dict.items() to iterate throught the keys and values simultaneously:

    ...
    for name, classes in teachers.items()
        if len(classes) > max_count:
            max_count = len(classes)
            name_of_teacher = name
    ...  

The task was only to return the name of the teacher with the most classes, but if you want it, you can return multiple values separated by commas, like this: return name_of_teacher, max_count. (You can "extract" them in the same order when you call the function.)

    ...
    return name_of_teacher, max_count

name, count = most_classes(teachers)
print("Teacher with most classes: {} ({} classes)".format(name, count))