According to Adam Smith, humans are self-interested individuals. If this is true, why does he argue that people should reinvest their wealth? How does this benefit a self-interested individual? Does morality play a role here? What role does productive and unproductive labor play here precisely?
As every individual, therefore, endeavours as much as he can both to employ his capital in the support of domestic industry, and so to direct that industry that its produce may be of the greatest value; every individual necessarily labours to render the annual revenue of the society as great as he can. He generally, indeed, neither intends to promote the public interest, nor knows how much he is promoting it. By preferring the support of domestic to that of foreign industry, he intends only his own security; and by directing that industry in such a manner as its produce may be of the greatest value, he intends only his own gain, and he is in this, as in many other cases, led by an invisible hand to promote an end which was no part of his intention. Nor is it always the worse for the society that it was no part of it. By pursuing his own interest he frequently promotes that of the society more effectually than when he really intends to promote it. I have never known much good done by those who affected to trade for the public good. It is an affectation, indeed, not very common among merchants, and very few words need be employed in dissuading them from it.
See also Adam Smith's Moral and Political Philosophy :
Smith is further from utilitarianism than Hume. [...] He believes that our faculties of moral evaluation are always directed toward the motivations and well-being of particular individuals in particular situations, not to goods that might be possessed jointly by groups of human beings, and he rejects the idea that our assessments or decisions should aim at the greatest happiness for the greatest number of people.
I think you are taking the term "self-interest" far too narrowly.
People may be motivated by all sorts of interests, and in rational agent theory, of which Smith is an ancestor, the "rational self-interest" may even extend to benevolent acts, if that is what the "self" pursues by rational means.
Smith's point was not that people always pursue or ought to pursue some narrow self-interest but, as in Mandeville's "Fable of the Bees," that the social sum of self-interested actions may have good outcomes overall. Thus the moral, traditional, or legal suppression of "gain" through self-interest was not necessarily justified on behalf of "king and country."
At a practical level, in any modern money economy and for the bourgeois of Smith's day, wealth will simply deteriorate if it is not invested. Even landed wealth. It was the new capitalist framework, the merchant traders, and the early "improving farmers," that made continual reinvestment a necessary component of wealth and, presumably, "self-interested" planning.
Moreover, since merchant capitalism entailed significant state involvement and collective investment in infrastructure and operating capital--docks, warehouses, ships, mail services, armies--the self-interested organization of "public-private" works was already well-understood.
Why should one suppose reinvestment be inimical to self-interest? Presumably when the investment matures it is I that benefits; I might, however, choose not to invest the capital if I had immediate need for capital - say in paying off a ransom, or paying a bribe; but this shows that self-interest surveys a range of conditions, and that too temporally to gauge what is in its own interest.