Why do I need 4-conductor cable to my generator?
This depends on whether your neutral-to-ground bonding is before or after the transfer switch.  If the transfer switch is your service entrance, you can bond neutral to ground there (on the common side, after the switch, inside the transfer switch enclosure), and do switching only for the hot conductors.  If neutral is bonded to ground before the transfer switch, then all points beyond the point of bonding are now 4-wire, and a transfer switch must switch both hots and also the neutral (not the ground wire).

You can't mix by having the utility power be one way (for example 4-wire) and the generator be another way (3-wire) because you can't switch the bonding.  It has to be either both 3-wire or both 4-wire coming up to the transfer switch.

Edit: since you have bonding at the entrance and at the generator, you need to switch the neutral in the transfer switch.  Because of that you need 4 wires from both sources in order to keep the neutral and ground separated so the two bondings are never connected together by more than one wire (it will be the ground wire).