When I run the application my menu screen shows, but when I click the screen to begin playing the game the game begins playing but the menu screen is still their overlaying the
Not sure but I think your render() method on GameScreen not called. you must implement method render(float delta) that use delta time for parameter.
render()
GameScreen
render(float delta)
replace
public void render() { // your code }
with
public void render(float delta) { // your code }