Using your carrier plan, Flyp works by placing and receiving calls through our system through what are called "bridge numbers". Basically, these bridge numbers connect users to our system and then our system connects them to whoever they are calling. So, your phone records will show inbound and outbound calls to our bridge numbers, but the actual number you are calling will not show up.
People who receive calls from your Flyp number will have your Flyp number show up on their bill as if it were any other number.
SMS text messages use Flyp's proprietary system and not your carrier plan so they will not appear on your phone bill.