Did they influence the war in any significant way?
Motorcycles were used in many ways. Most armies of ww2 used men on motorcycles to deliver orders when phone lines had been cut and radio contact was not possible (many armies lacked enough long range radio equipment to supply all units with it, especially early war). In older times, message riders on horses had been used the same way, but motorcycles were faster and more reliable.
Many armies used motorcycle troops for recoinnasance since they could often go along paths and roads where armoured cars and other vehicles could not. The early German Panzer Division had a motorcycle battalion for this purpose. The Italian Bersaglieri (literally sharpshooters) raised several battalions of motorcycle troops that were attached to motorised and armoured divisions.
These motorcycle infantry formations acted like the dragoons of old - they rode their motorcycles to the battle, dismounted and fought on foot. They were quick, light troops that still could pack some serious firepower and could go where other motorised troops often could not. They were used for recoinnasance, raiding and rushing ahead to take important points before the enemy could reinforce them, and hold them until other troops could catch up.
As such, they were instrumental in the early German offensives in Denmark, France, the Low Countries, Yugoslavia and the Soviet Union.