
The Officers are the instruments of Orsino's authority in the city — their arrest of Antonio in Act III is what separates him from Sebastian and drives the Act V confrontation, and First Officer's testimony in Act V provides the exposition that contextualises Antonio's presence in Illyria.
The Officers enter Act III, Scene 4 at precisely the moment when the play's plot threads are most tangled — the duel between Viola-as-Cesario and Sir Andrew has been interrupted by Antonio, who has drawn his sword in Cesario's defence, and Toby has drawn against Antonio. Into this chaos arrive the Officers, who cut through all of it with institutional efficiency: 'This is the man. Do thy office.' Antonio is under arrest on Orsino's warrant, and nothing else happening in the scene is their concern.
First Officer's lines are the most substantive. He identifies Antonio with certainty despite the sea-cap being absent, orders him taken, and when Antonio pleads his case — speaking of his devotion to Sebastian, accusing Cesario of ingratitude — First Officer cuts him short twice: 'What's that to us? The time goes by. Away!' and 'The man grows mad. Away with him.' His function is to make Antonio's situation irrevocable and to force the poignant confrontation with Viola that provides the audience its first real clue that Sebastian may still be alive.
In Act V, First Officer reappears to present Antonio to Orsino, and here his role expands slightly: he provides exposition, explaining who Antonio is — his piracies against Orsino's fleet, the ships he took, the leg Titus lost — and where he was found. His knowledge of Antonio's history marks him as a professional, someone who has been looking for this man. He then recedes as the principals take over.
The Officers are motivated by duty — they have a warrant, they execute it, they answer questions put to them about it. They are the impersonal machinery of the state briefly visible in a play otherwise occupied with private feeling.
First Officer is the more distinct of the two — his lines show someone who does his job with minimal fuss, cuts through emotional scenes with procedural authority, and knows the city's wanted men by face even without a hat. Second Officer is functional support.