Twitter offers "Cards" that will preview links that you share inside tweets. If the website that your link points to supports Twitter Cards, then the link will be displayed with a title, short summary of the content and a thumbnail.
Unfortunately, Instagram does not support Twitter Cards. If you share an Instagram image on Twitter, it will be shown as a completely plain link box in your feed.
WordPress Instagram Twitter Bridge is a WordPress plugin that lets you create links to Instagram posts that will show as nicely formatted Cards in your feed. It basically adds Twitter Card support to Instagram via your website. This can be helpful if you already happen to have a WordPress website and are using Instagram and Twitter as well.
Currently, the easiest way to install this plugin is to download it as a ZIP file from this repository and install it in your WordPress installation manually.
Once you've installed and activated this plugin, you will find the item "Instagram Twitter Bridge" in your settings menu in the WordPress admin panel.
There, you can configure the plugin and create "bridge" links by pasting the URL of a specific Instagram post. The link generated by the plugin can then be used to share on Twitter. If you share the link created by the plugin, it will be displayed with a thumbnail and a description in your feed.