Why do birds not get electrocuted on power lines?

Current flows in response to a difference in electrical potential, not merely to the presence of voltage. A bird perched on a single conductor has both feet at nearly identical potential, separated by a few centimeters of wire whose resistance is negligible compared with the bird's body. With essentially no voltage difference across it, almost no current passes through the bird, and it is unharmed regardless of how high the line's voltage is relative to the ground. The bird is not grounded, and grounding is what would complete the circuit.

The danger appears the moment a bird bridges two things at different potentials. Touching two separate conductors, or one conductor and a grounded metal structure such as a pole crossarm or transformer case, creates a path and current flows. This is precisely why large birds are electrocuted far more often than small ones: an eagle, hawk, or heron has a wingspan long enough to span the gap between a conductor and a ground, which a sparrow cannot. Raptor electrocution is a documented conservation problem, and utilities retrofit poles with wider spacing and insulating covers specifically to address it.

The same physics explains several human-facing rules. Line workers use bucket trucks and insulated equipment to avoid becoming the path to ground, and in some conditions work while bonded to the energized line at the same potential. If a power line falls on your car, staying inside is correct because the vehicle body keeps you at a single potential; if you must exit due to fire, the guidance is to jump clear without touching car and ground simultaneously, then shuffle away with feet together to avoid step potential, which is the voltage gradient across the ground near a downed line and which kills livestock in exactly that manner.