Learn Italian: Ieri il traffico era frenetico e mi ha reso irritabile. - Yesterday the traffic was frantic and made me irritable.

In Italian, when 'to make' means to cause someone or something to enter a new state or take on an adjective (like 'made me irritable'), you must use rendere. You cannot say mi ha fatto irritabile. The verb fare is used for causative actions with other verbs (like mi ha fatto piangere - made me cry), whereas rendere pairs with adjectives.

Both parts of this sentence happened yesterday. Why does it use the imperfect (era) then the passato prossimo (ha reso)?

The imperfect (era) sets the scene and describes a continuous state in the past: the traffic was frantic for a block of time. The passato prossimo (ha reso) is used for the specific resulting event or change of state: the traffic triggered a shift in your mood, turning you irritable.

Since mi is a direct object pronoun here, shouldn't the past participle reso change its ending to match my gender?

It can, but it is not strictly required. With the direct object pronouns mi, ti, ci, and vi, making the past participle agree is optional. If a woman is speaking, she could absolutely say mi ha resa irritabile, but leaving it as the default masculine reso is also perfectly correct and very common. Agreement is only mandatory for the third-person pronouns lo, la, li, and le.