I have been searching, what version of SharePoint 2013 I need to use Date Filter Web Part.
Can I enable it on Foundation or do I need Standard for this? Or is it Enterprise feature only?
I can see those in SharePoint 2013 Enterprise:
You are using SharePoint Foundation, that's why Date Filter web part is missing as unfortunately it only comes with the Enterprise edition of SharePoint.
If you still want to apply the date filter than there is a trick mentioned in below link which can help you to get desired output.