When the bird perches on a live wire, her body becomes charged--for the moment, it's at the same voltage as the wire. But no current flows into her body. A body is a poor conductor compared to copper wire, so there's no reason for electrons to take a detour through the bird. More importantly, electrons current flow from a region of high voltage to one of low voltage. The drifting current, in effect, ignores the bird.
But if a bird (or a power line worker) accidentally touches an electrical "ground" while in contact with the high-voltage wire, she completes an electrical circuit. A ground is a region of approximately zero voltage. The earth, and anything touching it that can conduct current, is the ground.
Like water flowing over a dam into a river, current surges through the bird (or person's) body on its way into the ground. Severe injury or death by electrocution is the result.
That's why a squirrel can run across an electrical line, but sadly die when its foot makes contact with the (grounded) transformer on the pole at wire's end.
It's also why drivers and passengers are warned to stay inside the car if it runs into a downed power line. Touching the ground with your foot would complete the circuit: Electrons would flow from the wire, into the car, and through you on their way into the earth. (Inside the car you are usually protected by the car's four rubber tires, which act as insulators between car and ground.)