Add the following code to the ThisDocument module of the document. Name the date content control and replace "Today's Date" in the code with the Title of the content control. Save as macro enabled document.
However, why not use instead a bookmarked date field which will always show the current date and doesn't need the macro?
Code:
Option Explicit
Private Sub Document_Open()
Dim objCC As ContentControl
Dim objDate As Date
For Each objCC In ActiveDocument.ContentControls
If objCC.Title = "Today's Date" Then
objDate = Date
objCC.Range.Text = objDate
Exit For
End If
Next objCC
End Sub