Why is $PV$ work negative?

Hint:Work is a term we generally use for the energy required to displace or change the position of the body against the applied force. To confine the discussion in thermodynamics: In a system, when we work on the system its internal energy increases but when the system does work on its surrounding the internal energy decreases.

Complete step by step answer:
The way the work is done by gases is: expansion and compression keeping the external pressure constant. Think what will happen if we increase the temperature of a gas which is in a container with a piston. The heat will increase the kinetic energy of the molecules of the gas due to which every time they collide with the piston, heat or energy of molecules are conveyed to the piston.

Due to the energy piston moves up and sees what happens: The work is done against the external pressure and volume of the gas increases. In this way the work is done by gas surrounding the external pressure. The equation of work done by gas:
\[w = - {P_{external}} \times \Delta V\]
Pressure external is the opposing pressure for the pressure inside the gas system.

And change in volume is calculated as:
\[\Delta V = {V_{final}} - {V_{initial}}\]
The system loses energy as volume of the system is increased and the work done by the system is negative. Thus, $PV$ work is negative.

Note:When we do work on the system the overall volume of the system decreases which results in positive work done.The scientific definition of work is different in many ways from its everyday meaning. The definition of work in physics reveals its relationship to energy – whenever work is done, energy is transferred.