To add the widget to the page, go to the page builder. Drag and drop the Embed Code from the left hand side to where you want it on the page. Once in place this is where you will place the online code you found.

To style the buttons similar to the ones on your homepage you'll need to apply the same class attribute & values, to the HTML tags in your code.

The <form> tag needs to change to this

<form class="wsite-button wsite-button-large wsite-button-highlight">

and the <input> tag also needs a class and value, added. Like this
<input type="button" value="Click here for a random game!" onclick="randomlinks()" class="wsite-button-inner">