building a twitter web app that only shows tweets containing links?

I know this is a big question but I need help figuring out this twitter tool.

So I have the android app called SmartNews, and on one of their custom tabs they allow you to login to your twitter to display a type of "news feed" from your timeline that is curated based on if a tweet has a link or not. I apologize if I'm failing to explain this well, but I will make sure to include photos to clarify what I am trying to recreate.

Here is an image of what the social tab looks like in the smartnews app. The requirements for this page are:

• user must login to twitter account to analyze timeline
• must be a link included in tweets from timeline
• tweets shown must be by those the user follows

If anyone could help me figure out how to duplicate this type of twitter tool so that it could be a standalone service it would be super helpful!

I don't have much experience with APIs and coding projects in general so any advice is welcome.