Let me first note that the voltage itself does not have any physical effect - the harm comes from the electric current. This happens in two ways:

• via the Joule heat when high current runs through the body (this requires currents of several Amperes, that we rarely encounter in the everyday life, but it is relevant for the bird)
• via triggering the heart fibrillation when the frequency of the alternate current is in resonance with the heart frequency, that is 50-60 Hz (colloquially this is known as electrocution)

Let us now look at the parallel circuit formed by the bird and the piece of the wire between its legs. The voltage, i.e. the potential difference between the bird's legs is definitely not the same as the potential difference between the wire and the ground (which is known to be a few kV). The current in the wire (a few Amperes) is partitioned between the bird and the piece of wire between its legs: $$i = i_{bird} + i_{wire}.$$ The potential difference between the bird's legs is $$V = i_{bird}R_{bird} = i_{wire}R_{wire}.$$ Solving these three equations we obtain: \begin{array} ii_{bird} = \frac{iR_{wire}}{R_{wire} + R_{bird}},\\ i_{wire} = \frac{iR_{bird}}{R_{wire} + R_{bird}},\\ V = \frac{iR_{wire}R_{bird}}{R_{wire} + R_{bird}}. \end{array} The resistance of a human body ranges from 1000 to 100000 Ohms, depending on whether it is wet or not - this could be a good estimate for the bird. The resistance of a copper wire is a few Ohms per thousand feet (depending on the wire diameter). That is the piece between the bird's legs has resistance of a few mOhms. Thus, $$\frac{i_{bird}}{i} = \frac{R_{wire}}{R_{wire} + R_{bird}} \approx \frac{R_{wire}}{R_{bird}} \ll 10^{-6},$$ i.e., the current flowing through the bird is a millionth part of the current in the wire or even smaller! It is too minuscule to cause any real damage. In other words: the piece of the wire between the bird's legs short-circuits the bird.

Let me first note that the voltage itself does not have any physical effect - the harm comes from the electric current. This happens in two ways:

• via the Joule heat when high current runs through the body (this requires currents of several Amperes, that we rarely encounter in the everyday life, but it is relevant for the bird)
• via triggering the heart fibrillation when the frequency of the alternate current is in resonance with the heart frequency, that is 50-60 Hz (colloquially this is known as electrocution)

Let us now look at the parallel circuit formed by the bird and the piece of the wire between its legs. The voltage, i.e. the potential difference between the bird's legs is definitely not the same as the potential difference between the wire and the ground (which is known to be a few kV). The current in the wire (a few Amperes) is partitioned between the bird and the piece of wire between its legs: $$i = i_{bird} + i_{wire}.$$ The potential difference between the bird's legs is $$V = i_{bird}R_{bird} = i_{wire}R_{wire}.$$ Solving these three equations we obtain: \begin{array} ii_{bird} = \frac{iR_{wire}}{R_{wire} + R_{bird}},\\ i_{wire} = \frac{iR_{bird}}{R_{wire} + R_{bird}},\\ V = \frac{iR_{wire}R_{bird}}{R_{wire} + R_{bird}}. \end{array} The resistance of a human body ranges from 1000 to 100000 Ohms, depending on whether it is wet or not - this could be a good estimate for the bird. The resistance of a copper wire is a few Ohms per thousand feet (depending on the wire diameter). That is the piece between the bird's legs has resistance of a few mOhms. Thus, $$\frac{i_{bird}}{i} = \frac{R_{wire}}{R_{wire} + R_{bird}} \approx \frac{R_{wire}}{R_{bird}} \ll 10^{-6},$$ i.e., the current flowing through the bird is a millionth part of the current in the wire or even smaller! It is too minuscule to cause any real damage. In other words: the piece of the wire between the bird's legs short-circuits the bird.

Remarks
I would like to return here to some aspects that are often overlooked when discussing electric circuits:

• It is possible (and very common) to have high potential difference (voltage) without any current flowing. Capacitors routinely accumulate voltages up to kilo- and mega-volts without a current flowing. Human skin has breakdown voltage of about 500V, that is a constant potential of a few hundred volts will not cause any current flow (and any harmful effects) at all! This is equally relevant for a bird. AC current presents greater danger, because ac impedance of a human body is much lower at frequencies of 50-60Hz.
• That voltage may exist without current is important to remember when applying the Joule's heat formula: $P = i^2 R$ and $P = V^2/R$ seem to tell the same thing, but both are applicable only when there is an actual current flow, not whenever a voltage is applied.
• There are no perfect sources of potential difference - connecting anything to a circuit changes the potentials and the currents in this circuit. In particular, one distinguishes voltage bias and current bias when talking respectively about the circuits designed to maintain the same level of bias or the same level of current. Fuses are used to detect exceedingly high current and prevent it from damaging the circuit (but interrupting the current flow). The bird in question finds itself as a part of a circuit where the current rather than the voltage is maintained.

Let me first note that the voltage itself does not have any physical effect - the harm comes from the electric current. This happens in two ways:

• via the Joule heat when high current runs through the body
• via triggering the heart defibrillation when the frequency of the alternate current is in resonance with the heart frequency, that is 50-60 Hz

Let us now look at the parallel circuit formed by the bird and the piece of the wire between its legs. The voltage, i.e. the potential difference between the bird's legs is definitely not the same as the potential difference between the wire and the ground (which is known to be a few kV). The current in the wire (a few Amperes) is partitioned between the bird and the piece of wire between its legs: $$i = i_{bird} + i_{wire}.$$ The potential difference between the bird's legs is $$V = i_{bird}R_{bird} = i_{wire}R_{wire}.$$ Solving these three equations we obtain: \begin{array} ii_{bird} = \frac{iR_{wire}}{R_{wire} + R_{bird}},\\ i_{wire} = \frac{iR_{bird}}{R_{wire} + R_{bird}},\\ V = \frac{iR_{wire}R_{bird}}{R_{wire} + R_{bird}}. \end{array} The resistance of a human body ranges from 1000 to 100000 Ohms, depending on whether it is wet or not - this could be a good estimate for the bird. The resistance of a copper wire is a few Ohms per thousand feet (depending on the wire diameter). That is the piece between the bird's legs has resistance of a few mOhms. Thus, $$\frac{i_{bird}}{i} = \frac{R_{wire}}{R_{wire} + R_{bird}} \approx \frac{R_{wire}}{R_{bird}} \ll 10^{-6},$$ i.e., the current flowing through the bird is a millionth part of the current in the wire or even smaller! It is too minuscule to cause any real damage. In other words: the piece of the wire between the bird's legs short-circuits the bird.

Let me first note that the voltage itself does not have any physical effect - the harm comes from the electric current. This happens in two ways:

• via the Joule heat when high current runs through the body (this requires currents of several Amperes, that we rarely encounter in the everyday life, but it is relevant for the bird)
• via triggering the heart fibrillation when the frequency of the alternate current is in resonance with the heart frequency, that is 50-60 Hz (colloquially this is known as electrocution)

Let us now look at the parallel circuit formed by the bird and the piece of the wire between its legs. The voltage, i.e. the potential difference between the bird's legs is definitely not the same as the potential difference between the wire and the ground (which is known to be a few kV). The current in the wire (a few Amperes) is partitioned between the bird and the piece of wire between its legs: $$i = i_{bird} + i_{wire}.$$ The potential difference between the bird's legs is $$V = i_{bird}R_{bird} = i_{wire}R_{wire}.$$ Solving these three equations we obtain: \begin{array} ii_{bird} = \frac{iR_{wire}}{R_{wire} + R_{bird}},\\ i_{wire} = \frac{iR_{bird}}{R_{wire} + R_{bird}},\\ V = \frac{iR_{wire}R_{bird}}{R_{wire} + R_{bird}}. \end{array} The resistance of a human body ranges from 1000 to 100000 Ohms, depending on whether it is wet or not - this could be a good estimate for the bird. The resistance of a copper wire is a few Ohms per thousand feet (depending on the wire diameter). That is the piece between the bird's legs has resistance of a few mOhms. Thus, $$\frac{i_{bird}}{i} = \frac{R_{wire}}{R_{wire} + R_{bird}} \approx \frac{R_{wire}}{R_{bird}} \ll 10^{-6},$$ i.e., the current flowing through the bird is a millionth part of the current in the wire or even smaller! It is too minuscule to cause any real damage. In other words: the piece of the wire between the bird's legs short-circuits the bird.