So you want constant $P,T$ while $V$ presumably changes due to the moving piston. But if $V$ is increasing (and if external $P\neq0$, i.e., not a so-called "throttling process"), then the gas is doing **work** on its surroundings. So if the process is adiabatic, then that work energy has to come from within the gas itself, whereby $P,T$ has to change (_a la_ first law) correspondingly. Otherwise, if you want the gas's $P,T$ constant while it's expanding, you'd have to supply outside energy to make up for the energy lost to the piston head, and the process won't be adiabatic.

**edit** Re your "what happens in particles?" part of the question, just consider a gas molecule colliding with the piston. If the piston's not moving, then, ideally, the collision's elastic, and macroscopic $P,V,T$ aren't affected. But in the non-equilibrium situation where the piston's expanding, some momentum's transferred from the molecule to the piston, resulting in corresponding changes to $P,V,T$.