what exactly happened on those parts of the front once these months long battles were over? Did these battles just fizzle out to where both sides just sat in their trenches from exhaustion?

In the case of the Somme, there was some small scale skirmishing here and there during the winter of 1916, but starting in February 1917 the Germans [fell back] (https://upload.wikimedia.org/wikipedia/commons/thumb/1/1e/Withdrawal_to_Hindenburg_Line_diagram_300px.jpg/220px-Withdrawal_to_Hindenburg_Line_diagram_300px.jpg) to the Siegfried Stellung, and the battlefield of the Somme became an area to the rear, until as Carlin notes the 1918 offensives. I doubt there were still corpses there, mass graves would have taken care of that, but the presence of old shell holes and trench lines did aid British 5th Army in it's fighting retreat in March, and to a lesser extent the German Army in August. In the case of Verdun, there was a series of major set-piece attacks launched in autumn-winter 1916, one in December being incredibly successful. There were more attacks by the French here, in 1917 following the mutinies and in 1918, in latter part of which the Germans abandoned the St. Mihiel salient, thus eliminating the 'right side' of the Verdun salient.

And by the sounds of this description, the battlefield seems empty. Why not have one side, such as the Germans in this offensive case, just punch through this spot that was so highly contested by the Allies only two years previous and now appears empty?

The battlefield was behind allied lines when the Germans attacked in March 1918. Moreover, the destruction left by the 1916 battle, the 1917 retreat, and the retreating British, greatly hampered the advance of the German forces during Operation Michael.

For more info on battlefield clean up, the FAQs section has a number of answers to previous questions regarding this.