By default the Trending #tags web part is added to your Newsfeed in MySites.
To use this web part outside of MySites you will first have to export it from the MySites’s landing page (e.g. http://server/my/default.aspx) as a DWP, then upload it to the page where you want to use it using the “Upload a Web Part” feature as shown below:
(I could not find the feature that would provision this web part – hence this export/upload workaround.)
The next thing you have to do is activate a feature to provision a required Display Template. The feature to activate is “My Site Host“. This is a hidden feature scoped at the Site Collection level. The feature ID is 49571CD1-B6A1-43a3-BF75-955ACC79C8D8. The feature folder under SharePointRoot is MySiteHost.
Failing to activate the feature above will cause the web part to error with the message below:
Display Error: The display template had an error. You can correct it by fixing the template or by changing the display template used in either the Web Part properties or Result Types.
Template ‘~sitecollection/_catalogs/masterpage/Display Templates/System/Control_TagFeed.js’ not found or has syntax errors. (LoadTemplate: )