How can I make the results of my website look like this on Google search?
I'm not really sure if this is possible in my website, because it's a one page website. If anyone could help and tell me if this is possible, and if it is how is it done.
The links under the main heading are called Sitelinks. They are automatically generated by google for some websites. Google Webmaster Tools allows you to demote sitelinks, but not to add them.
It wouldn't make sense to have Sitelinks to a one page website like yours. If you want them you'll have to add extra pages, and then hope that the google algorithms decide to link to some of them.