Vegans can eat unsweetened applesauce because unlike the sweetened applesauce the unsweetened applesauce does not contain trace amount of animal ingredients.
Some sweetened applesauce contains trace amounts of animal fat which means that the sweetened applesauce is not vegan.
Vegans do not eat any animals or meats so if they were to eat the applesauce that contained animal fat then they would be going against their Vegan diet.
The applesauce will usually say on the jar or can that it contains animal fat or not.
But the unsweetened applesauce does not contain animal fat and is vegan so unsweetened applesauce can be eaten by vegans.