0 represents all empty sets, i.e all sets with no elements in them. It is a value as is 1 or 2 or 3 (representing all sets with one, two or three elements in them). The natural numbers are based on set theory. Every natural number represents all sets with a specific finite number of elements, addition is combing the elements of both sets into one (which you can easily show is associative), multiplication is replacing each element in one set with all elements of the other (which you can easily show is commutative). 0 is the neutral element of this addition, i.e. adding it to anything leaves anything unchanged. This makes base-n systems for numbers possible. Those are representation of natural numbers that group elements in multiples of n + a remaining set with less than n elements in it. Which could possibly be the empty set. What n one choose as a base is just convention. But choosing a base is both natural and reasonable and makes calculation easy.

Are you puzzled by the fact that 0 times anything as well as anything times 0 is 0, although it doesn't look the same? You shouldn't. Math is about logic, not about how things may appear.