When they say "illegal" vehicles, they mean cars that were stolen from Israel. And the photo of one of the cars about the be destroyed shows an Israeli license plate:
The Palestinian Authority could arrange to return these cars to Israel. Almost certainly it costs more to destroy them and dispose of them than to transport them to a nearby checkpoint where Israeli companies could pick them up.
Instead the PA destroys them.
They gain nothing from destroying the cars.
But it seems that the idea of returning the cars to Israel is simply not considered. And knowing how Arabs in general adhere to a zero-sum game mentality, the reason is because anything that helps Israel must be bad for Palestinians in some way. Since returning the cars would make Israeli insurance companies and car owners happy, it is better to destroy them, because Israeli happiness means Palestinian misery, in this warped universe.
Now, think about how this twisted thought process applies to making peace between the two sides.