I believe there is a plugin called WP Lynx that does exactly that by crawling the page and generating a snippet that you can also edit. It's got a few minor bugs but generally it's a great plugin.
WP itself now supports the OEmbed, which means that you can grab snippets for a specific group of whitelisted websites. More info: https://codex.wordpress.org/Embeds#oEmbed
However, at the moment, not al lot of websites support OEmbed - much like it took some time before Facebook's OpenGraph metatags became widespread. The functionality should be more universal and better equipped to process any type of media, but it is still very much in development...