How do you see birds? Looking is how you see them. The better questions are how and where to look.
Have you ever looked for agates on a rocky beach? It takes a few minutes for your eyes to adjust to the appearance of agates. At first, you see little or no difference among the rocks. Then, an agate or two. Finally, eyes keyed to the task, agates are easier to see. It's the same for spotting birds.
With sufficient practice you create a search image: a mental image of what, generally speaking, a bird looks like, or what kind of movement defines a bird. Once acquired, this image can be yours for life.
The adjustment can be quick — not instant, but not years. If you look and listen, you can train your eyes to see and your brain to be quickly reactive.
Where to look
This all works better if the habitat is at least halfway appropriate. You might be surprised at what you find in your neighborhood, any neighborhood, if you just stand and listen. You will have more success, though, if you are somewhere with trees, bushes, shrubs, weeds, flowers, long grass, dirt, mud, puddles, and water small or large, standing or running.
Clue: Look for public land where the managing agency has created paths and trails for access. Paths and trails are a very good beginning.
Look in trees near and far, in tangles of shrub, in weedy patches, among reeds and cattails. Watch for movement. Look for things shaped like birds, and things that look odd. Watch for anomalies. Your search image is that anomaly.
Look at fences and wires and poles and posts. Hawks, owls and meadowlarks like to sit on posts. Smaller birds sit on wires.