High side driver internal design
In both cases they are talking of the worst-case conditions on the specified pins.  Because of the internal construction of the device, when the -150-volt pulse is applied to VCC or limp home power, the SEn pin can have -144 volts due to internal coupling to the VCC line.  If you have microcontroller outputs (labeled MCU in the diagram) driving these pins on the VN7016AJ, the VN7016AJ pins can pull down the microcontroller outputs down and damage the microcontroller.  You need to be sure that the microcontroller can supply enough current so that the voltage on this pin will stay in a safe range.  In their example calculation, they assume that the microcontroller can safely source 10 milliamperes.  When the pin is pulled low there will be nearly 150 volts (actually 144) across the 15K resistor, meaning that the part will have to source 10 mA.  If the resistor was only 10K, the part would have to source 15K, and so the output would be damaged from overcurrent or a negative voltage on the pin.

In every case, they are providing maximum expected voltage on each pin from normal operation of the part.  These values can increase for inductive loads.  Knowing this voltage, you must always choose a series resistor to limit the current injected into or supplied by your controlling part.