Once you have defined the plugin in your _config.yml, like you mentioned:
plugins:
- jekyll-stickyposts
As the documentation for jekyll-stickynotes states:
If you're not using a paginator at all, use something like:
{% assign sorted = site.posts | sort: "stickiness" %}
{% for post in sorted %}
...etc...
{% endfor %}
If you're using the original jekyll-paginate:
{% assign sorted = paginator.posts | sort: "stickiness" %}
{% for post in sorted %}
...etc...
{% endfor %}
Your site appears to be using the original pagination, so the second option suits you. The ...etc... is what you actually want to do with the sticky posts, the variable post can be used in the for loop and it will be equal to a stickied post from the sorted list which only contains sticky posts. From here, simply use HTML/CSS to style how you want the sticky to look at the top of the page. Keep in mind that this only sorts all your posts by stickied posts first, so you probably want to check to see if the post is a sticky post like so:
{% assign sorted = paginator.posts | sort: "stickiness" %}
{% for post in sorted %}
{% if post.sticky %}
<!-- Do something here with stickied post -->
{% endif %}
{% endfor %}
Put this code in your /_layouts/page.html to make it global across all pages, or /_layouts/post.html if you only want them to appear on top of posts.