Why have both: BJT and FET transistors on IC output?
Paragraph 2 of the description says:


  FAN3100 drivers incorporate MillerDriveTM architecture for the final output stage. This bipolar-MOSFET combination provides high peak current during the Miller plateau stage of the MOSFET turn-on / turn-off process to minimize switching loss, while providing rail-to-rail voltage swing and reverse current capability.


At the bottom of page 14 in the section *MillerDrive Gate Drive Technology" it goes on to explain:


  The purpose of the MillerDrive architecture is to speed up switching by providing the highest current during the Miller plateau region when the gate-drain capacitance of the MOSFET is being charged or dischared as part of the turn-on / turn-off precess. For applications that have zero voltage switching during the MOSFET turn-on or turn-off interval the driver supplies high peak current for fast switching even though the Miller plateau is not present. This situation often occurs in synchronous rectifier applications ecause the body diode is generally conducting before the MOSFET is switched on.


The answer to "Who can tell me about Miller Plateau?" explains it thus:


  When you look at the datasheet for a MOSFET, in the gate charge characteristic you will see a flat, horizontal portion. That is the so-called Miller plateau. When the device switches, the gate voltage is actually clamped to the plateau voltage and stays there until sufficient charge has been added/ removed for the device to switch.
  It is useful in estimating the driving requirements, because it tells you the voltage of the plateau and the required charge to switch the device. Thus, you can calculate the actual gate drive resistor, for a given switching time.


The BJTs are able to get the output moving while the MOSFETs are ramping up. The MOSFETS can then provide the rail to rail voltage swing.