PCA9685 board resistor on output
This is just basic protection from the outside world you need to interface to connecting to higher/lower voltages and thus pulling too much power from the IC's pins. Because the board is going to be connecting to unknown things, this is prudent. Imagine the device you connect the PWM line to has an error where the PWM line and ground pins are swapped. When the PWM IC drives 5 V on the PWM line, that would connect directly to ground, thus drawing a large current and possibly damaging the PWM IC.

If you are driving (small) MOSFETs directly and you are just using this schematic as a reference, then you may omit these resistors. However, if you are intending to connect anything you don't have full knowledge of, then include spaces for this very basic protection.

Pull down resistors are recommended at MOSFET gates if you cannot guarantee low driving impedance at any point. For example, if you are driving a MOSFET from a microcontroller, the output pins may be high-Z during reset and before configuration. Again, optional but good practice to at least leave a space.