How should I understand the intrinsic body diode inside a MOSFET?
The intrinsic body diode is the p-n junction between the body and the drain. In a discrete (standalone) MOSFET, the source and body are usually tied together for convenience to make a three-pin package. This means there's a diode between the source and drain:



If the source voltage is always lower than the drain voltage, the diode stays off, and everything works as expected. This means you can't (easily) use a MOSFET for switching a bidirectional signal. Discrete MOSFETs are almost always used for low-side switching, so this limitation isn't a big deal in practice.

You can see that the source and body are tied together in the standard schematic symbols for three-terminal MOSFETs.