In Eriador, the area known as the Trollshaws was actually a tract of land that stretched for nearly 50 miles east to west and was composed primarily of coniferous flora, although beech trees also appeared to be common.  It was the former holding of the Dunedain Kingdom (one of three) known as Rhudaur.  As Rhudaur began to decline, power was seized by a Hill-men Lord who was allied with the Witch-King.

In TA 1409, the remaining Rhudaurim were slain or scattered by the forces of the Witch-King of Angmar, and the area remained in his sway for over 500 years.  In 1975 at the defeat of the Witch-King, the men living in Rhudaur (who were not the Dunedain, but rather descendants of the Hill-men tribe that overthrew the kingdom 500 years previous, and were thoroughly corrupt) were themselves thrown down and utterly destroyed.

The area was virtually uninhabited, and eventually some trolls descended from the highlands of the Ettinmoors to the north and began dwelling there.  It was not uncommon for travellers to become waylaid by roving trolls preying upon them as they traversed the Great East Road, one of the few reliable paths through the region.  It was around May 2941 that Bilbo Baggins, Thorin Oakenshield and company were captured by three of these nefarious creatures; Tom, Bert and William (very likely not their original names.)  The three famously argued over how best to dispatch and consume the party of captives when their dispute carried them into the first rays of the morning light, transforming them into three massive statues within the glade where they resided.

It was in these trolls' discovered hoard that this fellowship happened upon a number of important artifacts that would come into play later in the history of Middle-earth.  Gandalf retrieved Glamdring while Thorin recovered Orcrist.  And Bilbo found a small blade he would eventually call "Sting." They buried the remainder of the trolls' ill-gotten treasure and placed a marked rock over the resting place.  Gandalf and Bilbo would return to recover the stashed loot on their return from their adventure with Smaug and the resulting conflict in 2942.

In 3018, Aragorn would lead a small band of frightened hobbits through the Trollshaws to escape pursuing Nazgul after a frightful encounter on Weathertop Hill, resulting in a greivous wound to Frodo's shoulder.  The trip was hazardous, as they stayed off the main road, and instead had to traverse felled trees, narrow ravines and dense thickets.  On 18 October 3018, the little troupe encountered the three stone troll figures, and Sam recited a poem that lifted the spirits of the fellows.  That afternoon, they found the Great East Road and continued their journey to Rivendell.  They had spent six full days making their way through the thickest part of the Trollshaws.

So there you have it...the Trollshaws were so named because the trolls came to dwell in parts of the woods after the defeat of the Witch-King.