The old phone system converts the sounds into a analog format, much like records.  The Internet phone service converts the sound into a digital format.  The digital format, like CDs and DVDs, allows for a better quality. 

When you speak into the phone, the internet phone device converts your voice into a digital packet.  This packet is then routed over the internet, the same way as email or web pages.  The digital packets are routed to a device that assembles the digital packets into standard voice.  The receiving device, connects to the local phone system and delivers the call to the answering phone.

The Internet Phone Service Provider pays only for a local call from the answering phone to the digital packet device that receives the call.  The rest of the call is carried over the Internet.