One of the indirect aspect (as opposed to the direct implications of poverty, like hunger or inadequate housing) is envy for things others have, perceived by Mathilde as unfairly distributed between women who have fine tastes, like herself, and those who do not appreciate what they have.  Further, Mathlide feels that she is unlike other women of her unfair status, who don't know or care about the better things that money can buy.   Maupassant writes:

Mathilde suffered ceaselessly, feeling herself born to enjoy all delicacies and all luxuries. She was distressed at the poverty of her dwelling, at the bareness of the walls, at the shabby chairs, the ugliness of the curtains. All those things, of which another woman of her rank would never even have been conscious, tortured her and made her angry. The sight of the little Breton peasant who did her humble housework aroused in her despairing regrets and bewildering dreams...She thought of long reception halls hung with ancient silk, of the dainty cabinets containing priceless curiosities and of the little coquettish perfumed reception rooms made for chatting at five o'clock with intimate friends, with men famous and sought after, whom all women envy and whose attention they all desire.

What is the indirect aspect of Poverty in the short story of "The Necklace" by Guy de Maupassant?

The direct aspects of poverty would be those that are immediate. For example, if you are poor, you don't have money. Often this means you must work harder, at less attractive work, and that you can't afford to take part in the same entertainments that others with more money could…or even the same necessities, as the poor have to skip medical treatments, live in cheaper houses, etc.

The indirect aspects are those that come as corollaries or side effects of these immediate aspects. For example, the wife in the story becomes less physically attractive as a result of the hard physical labor she has to perform. Because they are spending so much time working—and because they simply don't have the money for the appropriate clothing—the couple fall out of the same social circles. Technically, they wouldn't have to do so. A friend could still be a friend, but the couple becomes ashamed of what they've done, and of what they lack, and don't even try to spend time with their former friends.

As far as not using "I," that's pretty easy. Talk about what happens in the story directly, and cut your opinion out of it. Write, for example, "The wife wants to do…" not "I think the wife wants to do…"