To add at the bottom of page.list-collections.liquid. In the New Standard theme, use the content of page.full-width.liquid and copy the code at the bottom of that content. This won't work in the Lookbook, Brooklyn, Simple and Kickstand themes.

page.list-collections.liquid

{% comment %}
  Collections are listed here.
{% endcomment %}

{% capture uses_minimal_framework %}{% include 'product-loop' %}{% endcapture %}

{% if uses_minimal_framework contains 'Liquid error' %}
  {% assign uses_minimal_framework = false %}
  {% assign grid_item_width = 'large--one-quarter medium--one-third small--one-half large--col-3 medium--col-4 small--col-6' %}
{% else %}
  {% assign uses_minimal_framework = true %}
  {% assign grid_item_width = 'span3' %}
{% endif %}

{% assign image_size = 'large' %}

{% if linklists[page.handle].links.size > 0 %}
    
  {% assign number_of_links = 0 %}
    
  <div class="grid-uniform{% if uses_minimal_framework %} row{% endif %} clearfix">

    {% for link in linklists[page.handle].links %}

      {% if link.type == 'collection_link' %}

        {% comment %}
        If we have a collection link.
        {% endcomment %}   

        {% assign collection = link.object %}
        {% assign number_of_links = number_of_links | plus: 1 %}

        <div class="grid__item grid-item product-grid-item {{ grid_item_width }} text-center">
          <div class="grid__image product-grid-image">
            <a href="{{ link.url }}" class="grid-image--centered">
            {% comment %}
              Bring in the featured image of the first product in the collection if no collection
              image has been uploaded for it.
            {% endcomment %}
            {% if collection.image %}
              <img src="{{ collection | img_url: image_size }}" alt="{{ link.title | escape }}">
            {% else %}
              {% assign product = collection.products.first %}
              <img src="{{ product | img_url: image_size }}" alt="{{ link.title | escape }}">
            {% endif %}
            </a>
          </div>
          <p class="collection-grid__item-title">
            <a href="{{ link.url }}" class="collection-item__title">{{ link.title }}</a>
          </p>
        </div>
    
        {% if uses_minimal_framework %}
          {% cycle 'clear-item': '', '', '', '<div style="clear:both"></div>' %}
        {% endif %}

      {% elsif link.type == 'page_link' %}

        {% comment %}
        If we have a page link.
        {% endcomment %}

        {% assign number_of_links = number_of_links | plus: 1 %}

        {% assign linked_page = link.object %}                    
        {% assign have_image = false %}

        {% comment %}
          Does the page contain an image? If so, let's use it.
        {% endcomment %}

        {% if linked_page.content contains '<img' %}

          {% assign have_image = true %}
          {% assign src = linked_page.content | split: 'src="' %}
          {% assign src = src[1] | split: '"' | first %}
          {% capture image_suffix %}_{{ image_size }}.{% endcapture %}
          {% assign src = src | replace: '_small.', image_suffix | replace: '_compact.', image_suffix | replace: '_medium.', image_suffix | replace: '_large.', image_suffix | replace: '_grande.', image_suffix %}
        
        {% comment %}
          If the page contains no image, and that page features collections, let's grab the image of the first
          collection on it.
        {% endcomment %}

        {% elsif linklists[linked_page.handle].links.size > 0 and linklists[linked_page.handle].links.first.type == 'collection_link' %}

          {% assign have_image = true %}
          {% assign collection = linklists[linked_page.handle].links.first.object %}

          {% if collection.image %}
            {% assign src = collection | img_url: image_size %}
          {% else %}
            {% assign src = collection.products.first | img_url: image_size %}
          {% endif %}

        {% endif %}

        <div class="grid__item grid-item product-grid-item {{ grid_item_width }} text-center">
          <div class="grid-image product-grid-image">
            <a href="{{ link.url }}" class="grid-image--centered">
              {% if have_image %}
                <img src="{{ src }}" alt="{{ link.title | escape }}" />
              {% else %}
                <img src="{{ link.title | handle | append: '_' | append: image_size | append: '.png' | file_url }}" alt="{{ link.title | escape }}" />
              {% endif %}
            </a>
          </div>
          <p class="collection-grid__item-title">
            <a href="{{ link.url }}" class="collection-item__title">{{ link.title }}</a>
          </p>
        </div>
    
        {% if uses_minimal_framework %}
          {% cycle 'clear-item': '', '', '', '<div style="clear:both"></div>' %}
        {% endif %}

      {% endif %}

    {% endfor %}

  </div>

  {% if number_of_links == 0 %}

  <div class="grid">
    <div class="grid__item grid-item rte text-center">
      <p>
        There are no links of type <i>Collection</i> in your <b>{{ page.title }}</b> link list.
        Go ahead and <a href="/admin/link_lists/{{ linklists[page.handle].id }}" target="_blank">edit your link list</a> to use links that point to collections.
      </p>
    </div>
  </div>

  {% endif %}

{% else %}

<div class="grid">
  <div class="grid__item grid-item rte text-center">
    <p>
      You do not have a link list with a handle set to <kbd>{{ page.handle }}</kbd>, or you do and it's empty. 
      Go ahead and create a link list called <b>{{ page.title }}</b> on your <a href="/admin/links" target="_blank">Navigation page</a> and populate that link list with links that point to collections, to see collections listed here.
      Make sure the link list also has a handle set to <kbd>{{ page.handle }}</kbd>.
    </p>
  </div>
</div>

{% endif %}

<style>
  kbd { 
    background-color: #FBFBE4;
    padding: 1px 4px;
    border-radius: 3px;
  }
</style>

Replace:
{% assign grid_item_width = 'large--one-quarter medium--one-third small--one-half large--col-3 medium--col-4 small--col-6' %}
grid_item_width assignment to:
'small--one-half medium-up--one-third'

Is there a way to use this to display a single product? I have a few use cases which would benefit SEO if the page/blog post showed a single product instead of a collection (it seems silly to link to a collection that only contains one item!)

great code though, I will be using this for collections

I know this is old but in case anyone comes across the same problem like I've been trying to fix for hours:

I replaced the line (Line 9):
{% assign grid_item_width = 'large--one-quarter medium--one-third small--one-half large--col-3 medium--col-4 small--col-6' %}

With:
{% assign grid_item_width = 'col-xs-12 col-sm-4' %}

And it went perfectly into 3 columns rather then one single column.

@kurtmeister99 it worked for me but seems to go down ways, how do you change span