On-page links (or internal links) are links pointing from Page A (in this case your homepage) to any other page on your site.
Currently, your homepage is linking to many other pages within your site. But I think this is a normal practice. Take a look at any other popular site and you'll see the same thing (ex. http://www.huffingtonpost.com/). What I would suggest though, is to analyze user behavior and identify what links are being clicked on the most. If some of your links are never clicked on, is it really useful to have them there?
Howard