So what you are looking to do is post a link to a post or page from your site onto a social media site like Twitter, and you want an image and summary from your post to show up. That’s handled through a special metadata tags called Open Graph. While most social media sites uses Open Graph, Twitter has its own metadata tags called Twitter Cards (see here for the difference). For example, on that Psychology Today article that you used as an example on Twitter, if you go to the actual article and do a view source, you’ll see these Open Graph tags and Twitter cards:
<meta name="twitter:creator" content="@PsychToday" />
<meta property="og:title" content="Don't Ever Talk to Me Again" />
<meta name="twitter:description" content="Why having people cut out of your life can be detrimental to your health." />
<meta name="twitter:title" content="Don't Ever Talk to Me Again" />
<meta name="twitter:site" content="@PsychToday" />
<meta name="twitter:card" content="summary_large_image" />
<meta name="twitter:image" content="https://cdn.psychologytoday.com/sites/default/files/field_blog_entry_images/2019-08/breakup.png" />
<meta property="fb:app_id" content="220580041311284" />
<meta property="og:type" content="website" />
<meta property="og:app_id" content="220580041311284" />
<meta property="og:image" content="https://cdn.psychologytoday.com/sites/default/files/styles/image-article_inline_full/public/field_blog_entry_teaser_image/2019-08/breakup.png?itok=HjLzCXJ7" />
<meta property="og:description" content="Why having people cut out of your life can be detrimental to your health." />
<meta property="og:url" content="https://www.psychologytoday.com/blog/darwins-subterranean-world/201908/dont-ever-talk-me-again" />
So, for example, the Twitter card twitter:image tells Twitter what image to use on your tweet.
So now you know what to look for, you can search the WordPress plugin repository for either >Open Graph or Twitter cards. In most cases, if you set the featured image on your post, that’s what will be used as the image on your social media post.