How do I insert the Social Network icons at the intro section of the homepage?

The social icons are generated by the StagTools plugin, and you just need to insert the following shortcode in order to make them appear:

[stag_social]

This requires that you have already installed and activated the StagTools Plugin, and you have configured the Social Settings section under 
Settings > StagTools > Social

Then, at the page that you have defined to use as the intro section, just paste the shortcode and you should be good to go.