Click on the 'Balloons Nearby' tab at the bottom of the screen. On the map you will see a small red circle and a larger grey circle. These boundaries represent balloons near you that you can view.
Balloons in the red circle can be grabbed. Click on one of the red balloons to see a summary of its contents and click on that summary to see the full details of the balloon. On the top right of that balloon page you will see the 'Grab' button. Click on this to grab the balloon. Now you can add to the tale!
If you see a balloon you want to grab that is outside the red circle you'll have make your way towards it or hope it floats your way!