Learn Italian: I membri sospettosi agiscono per invidia contro di voi. - The suspicious members act out of envy against you.

In Italian, certain prepositions like contro (against), senza (without), dopo (after), sopra (above), and sotto (under) require the preposition di when they are directly followed by a personal pronoun (like me, te, lui, noi, voi, loro). Because of this rule, you must say contro di voi. If it were a regular noun instead of a pronoun, you would drop the di, as in contro il muro (against the wall).

Why do we use per in per invidia to mean 'out of envy'?

The preposition per is frequently used in Italian to express the cause, motive, or reason for an action. While English uses 'out of' or 'from' for emotions driving an action, Italian uses per. Other common examples include piangere per la gioia (to cry out of joy) or tremare per il freddo (to shiver from the cold).

Does sospettosi mean the members are suspicious of others, or that they look suspicious?

It means they feel suspicious of others. The adjective sospettoso describes a person who harbors suspicions, is paranoid, or is distrustful. If you want to say that someone or something looks suspicious and arouses suspicion in others, you would use the adjective sospetto, as in un pacco sospetto (a suspicious package).