What they typically do with Polymer is have two connected elements which shows only one and when you perform some action, the other gets shown (from display: none) and animates from certain dimentions to its final form. Sometimes elements also shift but it depends on whether the content is able to move to its new position or not.
You have to work with css transition, transform and display. Sometimes even custom animations. And you are mostly changing multiple divs to their final form. I think the most difficult would be animating colors (from white to pink or from yellow to green for example) as those are most difficult to do (performance-wise).
If you look at the example you've set (final link) you see there's a list of items with a detail div and once you click the item you show the detail and transform the contents to its final dimentions.
Just know that these things are pretty hard if you aren't very much into Angular or HTML/CSS/Javascript.
The framework of Polymer for Web is very much a work in progress and i wouldn't be surprised if it took a few months to get similar results for both native and web.
You can take example from things like this: https://medium.com/tictail-makers/giving-animations-life-8b20165224c5 or https://www.polymer-project.org/apps/topeka/ or http://codepen.io/collection/amheq/ . And don't forget to speed it up by using some bootstrap theme like this http://fezvrasta.github.io/bootstrap-material-design/ or something.
I've been struggling with the same problem as there isn't much to go from right now. You stated the Angular project but that will take time. If you want to do it now, you have to do quite some work (if you do, share it with us), but you might be better of with postponing this until most of the bugs and problems have been solved.
Thats what i'm doing now.
