Understanding CT sensors and current to voltage circuits
From what I understand the coil in the "sensor" is getting current
  induced by the AC phase I clamp it onto.


The current in the clamped AC circuit induces a voltage in the main coil (the secondary winding) of the CT. The current that flows in the secondary winding is only due to adding a burden resistor to the terminals of that coil.

Without a burden resistor, the voltage at the terminals of the secondary can be several tens of volts in amplitude because, in effect the primary is 1 turn and the secondary is (say) a thousand turns and although this makes a voltage magnification of 1000:1 the voltage at the primary is only a few several millivolts. That's the story without a burden resistor.

Just as the CT will magnify voltage by 1000:1 conservation of energy means that it reduces current into the secondary by 1000:1 but only when a burden resistor is added.

There are other subtleties related to how the burden (on the secondary) is transformed down into a very low impedance on the primary by the turns ratio squared and understanding this gives a real deep insight into the mechanisms at play but try and understand what I wrote above first.