Edit: Let me add a few words about the motivations. Once you already know the fundamental theorem of symmetric functions, the above isomorphism may seems tautological and not very interesting. In fact, the existence of an (actually several) very explicit isomorphism(s) from the algebra of symmetric functions to $A$ is nothing but a reformulation of this theorem. On the other hand, the above definition is arguably one of the most natural definition of $A$, and you get the Hopf structure for free.
Somehow, the fundamental theorem tells you that the algebra structure of symmetric functions is not that interesting. But it turns out that many interesting combinatorial identities can be deduced from the Hopf structure, and especially from the fact that it's self dual. Hence the pull pack of the coproduct and antipode to the algebra of symmetric functions itself has many interesting combinatorial applications. The same is true for the other Hopf algebra structure, since it identifies combinatorial identities between symmetric functions, and the computation of the composition inverse of a formal power serie.