I've been seeing more and more beautifully presented links popping up in my Twitter feed recently, so decided to research further. I do struggle with the limitations of Weebly's blogging features occasionally, but I managed to get this working well so thought I'd post in case there are any other Weebly users in a similar position.
A bit of quick digging around on Google revealed that these links are called Twitter Cards, and there's a full tutorial from the developers on that page. As usual, it's written for people who can access and edit the HTML for their page, but the fiddle to make this work for Weebly was surprisingly easy.
<meta name="twitter:site" content="@yourtwitterhandle">
<meta name="twitter:title" content="POST TITLE max 70 characters">
<meta name="twitter:description" content="POST DESCRIPTION">
<metaname="twitter:image:src" content="URL OF IMAGE">
It's also worth noting that the Twitter Card will not show up like this on summary feeds; it's only when your tweet is clicked and expanded that the full details are shown.