Assault rifles can shoot repeatedly and rapidly without reloading.
Nazi’s realized that in most fire-fights soldiers were less than 400 meters. While traditional machines guns could shoot considerably further, their range was unhelpful. Furthermore, the weight of both the weapons and ammunition became a liability.
In response, Nazi’s invented a light weapon with smaller and lighter ammunition. Their weapon combined a traditional rifle with a machine gun. It was light and easy to handle. After several prototypes, they settled on Sturmgewehr 44, the first mass-produced modern assault rifle, in 1943.
The Soviet Army a rifle to counter the German weapon. Their rifle needed to be light and fire rapidly but, given the large number of uneducated Russian troops, also easy to operate. After several earlier attempted, Soviet Mikhail Kalashnikov invented the AK-47, completed soon after WWII.