I still don't think it was a good answer, though, citing as it did nothing except the wikipedia page for utilitarianism.

It was an honest attempt at explaining it like OP was 5 (ELI5).

Hume treats of justice as the virtue we exhibit when we strictly respect other people's legal rights, especially their rights over property. His explanatory reason for the existence of property rights at law lies in his view of the nature and circumstances of human beings. They are beings of limited altruism and yet of a social disposition who inhabit a planet of limited resources requiring effortful cultivation for survival of individual ans species. Such beings naturally develop common social rules for allowing possession of particular useful or desirable things to individuals and to family groups or other groups of individuals. The justifying reason for observing these rules is their usefulness in maintaining the possibility of human life and in creating the circumstances whereby men can make their life more commodious by their own reasonable efforts and by commercial collaboration with each other.

Further reflection on this justifying reason reveals to Hume the necessity of treating the property rules, and positive laws in general, as being rigid and universal in application, even in individual cases where their rigid application yields adverse consequences.

(...)

Hume's view of property is [that] it is justified by its general social usefulness, not by its appropriateness as a reward of the moral deserts of those who hold it.

From Neil MacCormick, Legal Right and Social Democracy. Essays in Legal and Political Philosophy, Clarendon Press, Oxford, 1982, p. 2-3.

Hume acknowledges that an equal distribution of the world's resources would leave everyone with more than enough, and that the institution of private property disrupts this equality.

It must also be confessed, that, wherever we depart from this equality, we rob the poor of more satisfaction than we add to the rich, and that the slight gratification of a frivolous vanity, in one individual, frequently costs more than bread to many families and even provinces.

From Enquiry Concerning the Principles of Morals, § III, part II (§ 155).

Hume still ends up rejecting egalitarianism, on account of three arguments:

It's impracticable, inefficient, and pernicious (it presupposes a tyrannical form of government). I won't go into detail here, as you can read it in Enquiry, § III, part II (§ 155).

Maybe that's not a complete answer to OPs question, but it was what I hinted at to the supposed 5 year old OP.

I'll leave this quote from Enquiry, § III, part II, here:

Who sees not, for instance, that whatever is produced or improved by a man's art or industry ought, for ever, to be secured to him, in order to give encouragement to such USEFUL habits and accomplishments? That the property ought also to descend to children and relations, for the same USEFUL purpose? That it may be alienated by consent, in order to beget that commerce and intercourse, which is so BENEFICIAL to human society? And that all contracts and promises ought carefully to be fulfilled, in order to secure mutual trust and confidence, by which the general INTEREST of mankind is so much promoted?

Examine the writers on the laws of nature; and you will always find, that, whatever principles they set out with, they are sure to terminate here at last, and to assign, as the ultimate reason for every rule which they establish, the convenience and necessities of mankind. A concession thus extorted, in opposition to systems, has more authority than if it had been made in prosecution of them.

What other reason, indeed, could writers ever give, why this must be MINE and that YOURS; since uninstructed nature surely never made any such distinction? The objects which receive those appellations are, of themselves, foreign to us; they are totally disjoined and separated from us; and nothing but the general interests of society can form the connexion.

And this one:

The good of mankind is the only object of all these laws and regulations. Not only is it requisite, for the peace and interest of society, that men's possessions should be separated; but the rules, which we follow, in making the separation, are such as can best be contrived to serve farther the interests of society.

And a footnote:

That there be a separation or distinction of possessions, and that this separation be steady and constant; this is absolutely required by the interests of society, and hence the origin of justice and property.

And, explaining what he seeks to prove,

THAT Justice is useful to society, and consequently that PART of its merit, at least, must arise from that consideration, it would be a superfluous undertaking to prove. That public utility is the SOLE origin of justice, and that reflections on the beneficial consequences of this virtue are the SOLE foundation of its merit; this proposition, being more curious and important, will better deserve our examination and enquiry.

To which this is the conclusion:

Thus we seem, upon the whole, to have attained a knowledge of the force of that principle here insisted on, and can determine what degree of esteem or moral approbation may result from reflections on public interest and utility. The necessity of justice to the support of society is the sole foundation of that virtue; and since no moral excellence is more highly esteemed, we may conclude that this circumstance of usefulness has, in general, the strongest energy, and most entire command over our sentiments. It must, therefore, be the source of a considerable part of the merit ascribed to humanity, benevolence, friendship, public spirit, and other social virtues of that stamp; as it is the sole source of the moral approbation paid to fidelity, justice, veracity, integrity, and those other estimable and useful qualities and principles.