Probably the same way many other models are trained: autoregression or autoencoding. You'd either predict the next symbol, or you compress it into a latent space and reconstruct the original. My assumption is that birdsong is sequential, but this isn't something I know about. An entire song might be the smallest semantic unit, like the alien language in Arrival, though I think this is unlike.

Sometimes birds communicate via melody (songbirds), sometimes via repetition (ducks, geese), sometimes via timbre (pigeons, doves), and sometimes via a combination of the former and sound-making in general.

Sometimes birds communicate nonverbally as well, as in gaze direction and body posture.