How do get my sitelinks to appear on Google search results?
There is no way to do this. Google decides whether to show sitelinks or not.
that's not a sitemap, it's sitelinks. Even after you get your site into google, sitemap or otherwise, you won't get site links unless they deem that it's appropriate for your site. Google webmaster tools is a good place to start, but don't expect site links any time soon.
Per your picture, you are referring to sitelinks and not sitemaps in the Google jargon. Here is what Google Webmaster site has to say about sitelinks:
Google has not generated any sitelinks for your site. Sitelinks are completely automated, and we show them only if we think they'll be useful to the user. If your site's structure doesn't allow our algorithms to find good sitelinks, or we don't think that the sitelinks are relevant to the user's query, we won't show them. However, we are always working to improve how we find and display sitelinks.
You can get sitelinks in Google's search results, but takes time. Suppose if you want to get your product page under site links, you need to promote it and see that you get lot of hits to it, the page should be user and search engine friendly. It should appears on top pages with your target keywords. Now its up to you how you promote it online, once you get good hits to the page with top ranking you can get the results in couple of months.
What ever I suggest above as per my experience, you can try but I can't guarantee it'll work.
What you're seeing are "sitelinks" which aren't generated from your sitemap. Google don't give details of where these links come from, but they appear to be related to things like a clear navigation structure.