Cut away the bottom vines a few inches from the ground.
Pull as much of the vines away from the trellis or wall as possible. As the vine dies the leaves will dry and the rest of the vine will come free. Or, you can take the time to pull each section free.
Use a spade to dig up the roots of the trumpet creeper. Be sure not to damage the roots of any neighboring trees. Remove any suckers and the roots.
Observe the area occasionally to ensure all the roots and suckers were removed. Any remaining pieces can quickly re-root and become new trumpet creepers.