Among the many benefits of paying attention to bird sounds is that they give you an insight into what the birds are doing. Through their songs and calls the birds announce not only their presence, but also what they are doing.
One example is the mobbing of predators. Birds give alarm calls when they see a predator, and in many cases they even have calls specific to an avian predator like a hawk or owl. In June 2011 I was at the Pine Butte Guest Ranch in Montana, and heard alarm calls from several species including some very agitated American Robins. By carefully tracking the sounds up the hill through the forest I was eventually able to find and digiscope this Northern Pygmy-Owl, sitting at the base of a big Douglas-Fir with a Red Crossbill in its talons.
The robins led me right to it (which was probably their intention), but only because I understood their calls and thought that I might find a predator there. This is something you can learn through experience, and you can learn more quickly just by paying attention to these calls. The next time you see a hawk or an owl, or if you’ve attracted some songbirds by using an owl call imitation, listen carefully to the sounds they are making, and then listen for the same sounds later while you’re birding.
Birds give a lot of “false alarms” or brief low-level alarms. With practice you will become sensitive to the higher intensity of real alarm calls, and when these calls are sustained for several minutes, and directed at one spot, you can be fairly certain a predator is there. If you hear robins that sound like the ones in this video, go check it out!