Did you notice how Imogen is called a bird? Iachimo calls her "th' Arabian bird" when he visits her (1.6.20), and Arviragus announces her death by saying "the bird is dead" (4.2.197). What gives?
Well, the Arabian bird is a phoenix. Since traditionally, only one phoenix can exist at a time, Imogen's association with one highlights how rare she is. She lives in a world filled with lies and mischievous plots, but she never schemes; she's virtuous almost to a fault. But she also has something else in common with the phoenix: she rises from the ashes. Maybe this association foreshadows her death and resurrection to come.
So, if Imogen is a rare, magical bird, what does that make everyone else? Well, Belarius tells his companions to "perceive me like a crow" (3.3.14). It seems like Arviragus and Guiderius take this to mean he's intelligent, but it could hint at his manipulative nature, as well. Crows are known for both for intelligence and for darker qualities, after all.
When Imogen tells her father she doesn't regret marrying Posthumus over Cloten, she uses bird imagery to describe her feelings. She declares, "I chose an eagle, and did avoid a puttock" (1.1.167-168). A puttock is a greedy scavenger, which sums up Cloten pretty well. And a strong, noble, intelligent eagle is a fitting symbol for Posthumus.
By the end, we've got a bunch of birds flying around the play. But these aren't just any birds: each one symbolizes a character's personality, and each one gives us clues about that character's past or future.
So it should come as no surprise that birds are a central part of the oracle at the end of the play: not only does Jupiter soar like an eagle (or on an eagle to be precise), he also leaves a cryptic message for Posthumus by means of birds. The dream here seems to be an augury—that's a way of telling the future by watching the behavior of birds.
When the Soothsayer interprets the oracle, she states: "For the Roman eagle, / From south to west on wing soaring aloft, / Lessened herself and in the beams o' th' sun / So vanished; which foreshadowed our princely eagle, / Th' imperial Caesar, should again unite / His favor with the radiant Cymbeline" (5.5.570-574). Instead of just saying Rome and Britain are pals again, she uses "eagle" to represent something Roman: princely, noble, powerful, smart—you get the idea.
But even something as powerful and majestic as the eagle is no match for the sun. Eventually, the eagle (Rome) disappears into the sun (Britain). Hint, hint: Britain is better and more powerful than Rome.