"Getting Started" tiles are new addition to SharePoint 2013 team sites. These tiles intend to provide shortcuts to common actions, such as sharing your site, applying a theme. In SharePoint 2013, They replaces the “Getting Started” links from its previous version: SharePoint 2010.
From the home page you can remove this web part by simply clicking on "REMOVE THIS" link as shown in the above image. But it remains in the gallery in the event you would like to have it added back to the site. Just edit the page >> Insert >> Web Part >> Choose "Getting Started with your site" under "Media and Content" group, to add it back to your home page.
You can also find "Getting Started" link which is pointing to "GettingStarted.aspx" page under site action menu in SharePoint 2013.
Want to customize get started web part in SharePoint 2013? You can create new promoted tiles web part add to home page. Here is how: Create Custom Promoted Link Tiles in SharePoint 2013
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