You haven't given us too much to go on (what searches are you performing and what results are being delivered - are they local results etc.) However...
The first step to replicating (and surpassing) a site's success is to understand it.
To get you started, take a look at Moz's Open Site Explorer, searching around this website and you'll see that this site has only a small number of established links from a handful root domains, though the ones present almost all relate to Wedding Photography, none of them are spammy and most have a respectable Domain Authority. If you look Linking Domains and change Target to Root Domain (https://moz.com/researchtools/ose/domains?site=emilyhelen.com&target=domain&sort=domain_authority&page=1) you'll see a list of these, the linking pages and their DA.
You can also Compare Link Metrics for your site against this, here: https://moz.com/researchtools/ose/comparisons?site=emilyhelen.com
Quality, inbound links which pass on link equity from related sites are a big factor as they serve as endorsements, so you'll certainly want to develop as many of these as you can and the best way to do this is through building relationships with the humans behind the sites and getting them to engage with your content - if they like it there's a good chance they'd be happy to share it with their audience.
Also, be sure to check that your assumptions are correct. To do so, take advantage of the Rank Checker, here: https://moz.com/researchtools/rank-tracker which you can use to confirm your (and your competitor's ranking) for a variety of keywords, to check that what your seeing in your browser is a true reflection of what the rest of the world are seeing.
Beyond that, this site has nothing much in the way of text content (that I can discern at a glance).
I hope that helps give you a clearer understanding.
Good luck!