Simple ttk ComboBox demo

Just bind the virtual event <<ComboboxSelected>> to the Combobox widget:

class App:
    def __init__(self, parent):
        self.parent = parent
        self.value_of_combo = 'X'
        self.combo()

    def newselection(self, event):
        self.value_of_combo = self.box.get()
        print(self.value_of_combo)

    def combo(self):
        self.box_value = StringVar()
        self.box = ttk.Combobox(self.parent, textvariable=self.box_value)
        self.box.bind("<<ComboboxSelected>>", self.newselection)
        # ...

In the more general case, if you need to get the value of a Variable when it is updated, it would be advisable to use the tracing facility built into them.

var = StringVar()  # create a var object

# define the callback
def tracer(name, idontknow, mode):
    # I cannot find the arguments sent to the callback documented
    # anywhere, or how to really use them.  I simply ignore
    # the arguments, and use the invocation of the callback
    # as the only api to tracing
    print var.get()

var.trace('w', tracer)
# 'w' in this case, is the 'mode', one of 'r'
# for reading and 'w' for writing

var.set('Foo')  # manually update the var...

# 'Foo' is printed