set the first textbox's binding mode to TwoWay and UpdateSourceTrigger to PropertyChanged.
Binding Mode TwoWay makes the View communicate to and fro with the property in ViewModel. Setting UpdateSourceTrigger to PropertyChanged makes changes are communicated whenever the value in the textbox changes.
also, make a habit to just use / at the end of controls that you dont plan to customize much.
So it becomes like this:
<TextBox Grid.Column="0" Text="{Binding Str, Mode=TwoWay, UpdateSourceTrigger=PropertyChanged}" />