If you want to hide it from the view model logic, you can expose a property that indicates if the menu should be visible. Then you can bind that property in the xaml to the Visibility of the menu and set it to Collapsed.
<Window ...>
<Window.Resources>
<BooleanToVisibilityConverter x:Key="visConverter"/>
...
</Window.Resources>
<Grid x:Name="LayoutRoot" Visibility="{Binding ShowMenu, Converter={StaticResource visConverter}}">
...
</Grid>
</Window>
Another option is to use a trigger in the button to launch an animation that hides the menu:
<Window ...>
<Window.Resources>
<Storyboard x:Key="HideMenuAnim">
...
<ObjectAnimationUsingKeyFrames Storyboard.TargetProperty="Visibility" Storyboard.TargetName="LayoutRoot">
<DiscreteObjectKeyFrame KeyTime="0:0:1" Value="{x:Static Visibility.Collapsed}"/>
</ObjectAnimationUsingKeyFrames>
</Storyboard>
...
</Window.Resources>
<Grid x:Name="LayoutRoot">
...
<Button x:Name="StartButton"...>
<b:Interaction.Triggers>
<b:EventTrigger EventName="Click">
<b:ControlStoryboardAction Storyboard="{StaticResource HideMenuAnim}" ControlStoryboardOption="Play"/>
</b:EventTrigger>
</b:Interaction.Triggers>
</Button>
</Grid>
</Window>
Is this what you are looking for?