You need to activate the feature before it becomes available.
Go to Settings -> Site Settings -> Manage site features and look for Offline Synchronization for External Lists and click activate.
Now go to the list/webpart again and you should see the button enabled.
as noted by oc3lot you need exchange, no need for outlook or any other office installed on the server!
1.Go to the SharePoint site and browse to the page where you will integrate the calendar.
2.Click Page and Edit Page.
3.Click Add a Web Part. Click the Outlook Web App category.
4.Click the My Calendar Web Part, and click Add.
5.Click "Open the tool pane" on the My Calendar Web Part.
6.Enter https://mysite.outlook.com in the Mail server address field.
7.Enter the email address for the account's Exchange calendar. Click OK.
to know what your mail server is: