I think maybe I missed the explanation during the tutorial, but I'm not entirely sure what choosing streaks does, or how it is supposed to work.  

A written out explanation would help my brain :3

Picking the "streak" option is only available when you killed all enemies in a wave. When this happens, you can gain bonus points for the end of the run (helps toward earning stars). And they stack as long as you don't interrupt the streak by failing to clear a run or picking one of the other rewards instead. Basically, if you keep picking the streak options it ramps up your score more, which in turn gives you more stars toward unlocking things.

Woo!  Thank you!  I wasn't sure how they worked in regards to the run.