Jetpack itself doesn't control the way your tweets appear on Twitter.
When you publish a new post and when a link is posted to Twitter, Twitter crawls the post and looks for meta data that will give more details about your post: an image, a description, a title, ...
You can add this meta data to your theme manually, by following the instructions here:
https://dev.twitter.com/docs/cards
You can also use plugins that will do this for you. Here is an example:
http://wordpress.org/plugins/twitter-cards/
I hope this helps.