Symbolism, Imagery, Allegory
Just before Troilus and Cressida are separated after their first night together, they exchange love tokens. (Aw. How sweet!) Cressida gives Troilus her glove and Troilus gives Cressida his sleeve as the two lovebirds promise not to cheat on each other (4.4.69-71). And, no, the sleeve isn't still attached to Troilus' shirt, but it's probably really fancy and has lots of embroidered embellishments.
This is a chivalry thing, kids. Knights often wore their ladies' "favors" (a.k.a. scarves, veils, handkerchiefs, etc.) when they jousted or went into battle. And, yeah, we know this play technically goes down in ancient Troy, but Shakespeare is writing Troilus like a throwback to those chivalric medieval knights we've all read about.
So, the love tokens are supposed to symbolize the couple's love and commitment to one another, right? But, of course, Cressida betrays Troilus about a nanosecond later when she promises to become Diomedes' lover. Just in case we don't get what a traitor she is, Shakespeare has her give Troilus' sleeve to her new man… while Troilus watches from a hiding spot (5.2.66). Ouch.
Not only that, but Diomedes brags that he's going to wear the sleeve on his helmet the next day in battle just to taunt the guy who gave it to Cressida in the first place (5.2.92-93). Double ouch.
In the end, the sleeve becomes a big, glaring symbol of Cressida's sexual infidelity. Does this sound kind of familiar? Shakespeare does something similar with the infamous handkerchief in Othello. Othello gives Desdemona his handkerchief as a symbol of his love, which then gets stolen and winds up in the possession of another man. Well, Othello sees the handkerchief as evidence that his wife's a cheater, even though she is most definitely not.
Here, though? The girl's totally cheating. There's a perfect match between the thing (the sleeve) and the thing it symbolizes (the infidelity).