When Morpheus is taking Neo on a tour of the ship, he points out the core.
This is the core where we broadcast our pirate signal and hack into the Matrix.
He's not just talking about cracking a firewall to upload people and letting their minds jump over. When one of the crew is inserted into the matrix, it's a constantly running signal, or their bodies wouldn't be constantly jumping every time the Matrix is giving their avatars physical trauma. The pods in the power plant have a constant connection; their ship signal is faking the same effect.
This is also why they are hesitant about using the EMP once they are under attack at the end of the movie. If they use it, they break Neo's link, and Neo dies.
Regarding the phone, well, the phone is slightly symbolic as a connection to somewhere else. More practically though, disconnecting someone safely from the Matrix is a major chore under normal circumstances, since machines don't let a battery go until it dies (witness the setup that gets Neo out the first time). So tapping into the phone system as part of their hack to provide a safe disconnect point is just part of their mission run.