Current flows in a loop (which means the circuit is closed). A bird sitting on a transmission line does not complete the circuit. If the same bird keeps one leg on one track and another leg (or any part of its body) on another line(or the neutral points), then it will get roasted.
Consider this circuit where a bird sits on the wire:
The values R1 and R2 are resistances of the line. Electricity takes the path of least resistance. The two legs of the bird which are perched on the same line do not complete the circuit. The R_Bird( resistance of the bird’s body) is much higher than that of the line, so the bird might not experience high current. The potential difference between the two legs of the bird is the same( since the resistance of the line is the same throughout). The current flows on. The bird is safe.
Now consider this scenario:
A bird sitting on a line decides to fly away and raises the wings. With one wing touching the neighboring line and the leg on the first line, this creates a closed circuit. Thus electricity (following the path of least resistance) will detect a potential difference between the wing (which touches the other line) and the leg(which is placed on the first line). The current tries to take on that path creating a short-circuit. Eventually, the bird gets zapped and falls off the line. Now the current will continue to flow on.
