You've got your camera set up, the settings are dialed in, your lens is fitted, and that Northern Mockingbird has finally ventured closer than he's ever approached you before. In the birding hobby it's go-time. You have the opportunity to capture an image that could be magazine quality and this is no time to choke. You glance at your settings and push the shutter release half way down. Before your eyes, he disappears. Ah, the old disappearing bird trick...
How do you stop scaring birds away? By understanding the birds and understanding their fight-or-flight triggers. Birds are under constant threat, from that alley cat who lives down the street to the owl that lives in the barn down the road, life isn't easy for a bird and they are very vigilant. They have to be. The difference between living to tweet another day or being a first class second course can be very close.
Birds are easily scared (also called "spooked") by many things but the biggest pair of triggers are sound and motion. The way you manage your camera strap can scare a bird away if it happens to have a threatening color, or if it moves unexpectedly. Your shutter noise can spook a bird and put him on high alert. Their wariness is the number one reason a lot of birders give up: you have to learn to present yourself as non-threatening and as just another part of the landscape.
First, minimize as much camera noise as you can. The shutter noise of a DSLR can sometimes be turned down via the settings but cameras have other noises, like the beep that accompanies autofocus lock. This can be disabled and it should be. Never assume a sound means nothing to a bird. For all you know, that innocuous beep may sound like a beep that is emitted from the tracking collar of that cat down the street and, to the bird, it's an immediate sign of peril and danger. Turn off your cell phone! The shrill squeal of a ringtone can sound like a predatory bird honing in for the kill.
Secondly, minimize unnecessary motion. Your green camera strap swaying in the breeze, to you it's a green camera strap. To a small bird in a big world it's a grass snake swaying before a strike and is something to fly away from with all possible speed. The clothing you wear can make you move more than you need to because it's uncomfortable and needs adjustment. Wear clothing that is comfortable so you're not constantly battling pants that are too tight and binding, or a shirt that constricts your arm movements.
Third, never leave home without your cough suppressant and allergy meds. For this birder, it's peppermint hard candy. For some it's an ordinary cough drop. When you're outdoors you're in the midst of all sorts of allergens from pollen and ragweed to freshly mowed lawns and dust. A sneeze followed by a few dozen coughs is an excellent method for scaring birds away and works only slightly less effectively than small arms fire. You need to find a way to suppress that as much as you can.
Try to become part of the environment. Calm colored clothing, quietly and patiently waiting, and trying to be a non-invasive as possible. If you want to stop scaring birds away you need to minimize the impact your presence has on the environment. If you're a naturally fidgety person, bring along a coin to manipulate while you wait. The biggest "trick" is to figure out what you're doing and then figure out how to stop doing that or do that differently so you don't spook them every time one ventures close enough for a great photo.
A final word of advice: don't worry so much. You will spook a few birds. Actually, you'll spook a lot of birds. Experience and time will help you learn "what the birds will tolerate" and help you improve. Even if you only get one "okay" picture of a house sparrow, a blurry shot of a mocky flying through, and one image of a house wren studying you, you still had a better day than the guy who's paving a road in the hot summer sun. Enjoy the experience and don't pressure yourself to be so perfect. This is a hobby, enjoy it. If you spook a bird, it's not the end of the world because you know one more thing you shouldn't do. It's only a failure if you make a mistake and learn nothing from it.