Trench warfare relied on using fixed positions to create a "no man's land" that was impassible to enemy forces. Between WW1 and WW2 there were vast improvements in the reliability of armored vehicles and the amount of power that could be generated by engines, leading to planes that could fly troops across no man's land to parachute down and harass enemy lines from the rear and tanks that could operate as a stand alone mobile armored division instead of being tied to infantry battalions due to their reliance on infantry support like the slow fragile tanks of WW1.

That was one of the key advantages that allowed German forces to roll over their neighbors the way they did. French forces still viewed infantry as the cornerstone of their forces and they spread their tanks out evenly among their infantry divisions in a support role, while Germans kept their tanks separate to make use of their mobility. They would use blitzkrieg tactics to punch holes in enemy lines, sometimes as narrow as a single roadway, and as their tanks moved deeper into enemy territory the infantry would follow behind, then all would turn and hit the lines from behind. French forces were unable to respond because the combined tank/infantry divisions suffered being no faster than the slowest unit and without the armor to slug it out with a German division focused on tank warfare. They just weren't ready for a fluid battlefield.

And there were other factors adding to this, knowing they were about to start a war and knowing they would be on the move, German's had radios in every tank so they could communicate more quickly. The communication between tanks was a huge leap and really cleared the fog of war and allowed the coordination of forces in a way no one had ever done before and no one on the other side was really prepared for. Basically it all came down to Germans being prepared for mobile warfare and setting the pace for every engagement by staying on the offensive.

Body count is a good question. You would think that since men were no longer stuck sitting in trenches being bombarded by artillery that casualties would have been fewer, but considering that the allies ultimately fought Germany to the bitter end I would say that any military age male who was going to die in a trench probably ended up fighting and dying somewhere else. Civilian casualties were probably greater, however, as the lines moved first across France and then back across Germany as tide of war turned.