Not all card readers are created equal, nor are they compatible with the Payment app.
If you did not order the card reader via https://paymentforstripe.com/order(or directly from the Stripe Hardware Shop), it will not work with the app.
When ordering the card reader, make sure to order it from the official link above. For security purposes, card reader companies (like Magtek and BBPOS) add app specific keys to all of their card readers, so ordering from somewhere like Amazon will NOT work with the Payment app.
Below, you will find a link to the two supported card readers for iOS and Android: