Why is neutral wire connected to ground at the transformer?
I understand the use of ground wire at the home appliances but ...


The earthing / grounding of applicances helps in two ways:


It prevents the appliance case or chassis getting a high potential with respect to ground. Without this protection a live appliance (due to internal fault, for example) would present a risk to life should a person touch the appliance and have sufficient path to earth for a few tens of milliamps to flow.
It provides a low impedance return to the transformer and when sufficient current flows it will blow the fuse or trip the breaker.



  ... why is the neutral wire connected to ground at the transformer? 


Connecting to ground at the transformer (or at the incoming connection point, depending on local regulations) ties the return conductor to ground and effectively "neutralises" it. Because it presents a low risk of significant voltage on it the neutral lines are normally unfused.


  Why doesn't the neutral wire go back to the power generation plants.


The diagram you provided hints at this. 


There is no connection between the three-phase, high voltage primary and the low-voltage secondary. 
The supply is feeding a three-phase transformer without a neutral on the primary side.
The high-voltage network may be "floating" with no direct earth reference. This means that the distribution system can sustain a single earth fault on any one of its phases without causing an unplanned power interruption. This wouldn't be possible if the distribution system used a neutral as well.




From the comments:


  I didn't understand the part about low impedance return path in the first part ...


Imagine that we have the option of earthing the neutral locally or back at the power station. The short local cable might have a resistance of, say, 0.05 Ω to earth while the much longer cable back to the substation might be, say, 10 Ω. Now create an earth fault by touching a live wire to the metal case of an appliance. Let's say that 10 A flows to earth. What voltage will the case rise to?


For the local earth \$ V = IR = 10 \times 0.05 = 50\ \text{mV} \$. This is very safe.
For the power station earth \$ V = IR = 10 \times 10 = 100\ \text{V} \$. This is dangerous.


The local neutral-earth link is safer.


  ... and the neutralising part of the 2nd highlight.


To "neutralise" means to make something ineffective. To neutralise a current carrying cable means to remove its voltage or potential difference with respect to earth. We do this by earthing it. In your picture we now have four current carrying conductors, three of which have high voltage with respect to ground and one, the neutral, will have close to zero potential as it has been neutralised.




  So in order to minimize the potential of the appliance case in case of faults we need to choose the low impedance return path. 


Correct.


  'return' - does it mean that the ground wire actually is a part of a loop (as if connect to the power station at some point). 


No. The transformer is isolating. There is no connection between the primary and secondary so no current flows from the house back to the power station. As far as the house is concerned the local transformer is the "power station".


  Ok, I now understand neutralising. So earthing an appliance is also neutralising. Isn't it?


No, that's not quite the right way to think about it. There is normally no potential on the chassis or case of the appliance. They are not conductors. But you are correct that it prevents the chassis / case from achieving a high voltage.




  So only a conductor which is always at a potential can be neutralised?


That doesn't make sense. If it's always at a potential then it can't be neutralised. Only if the supply would otherwise be floating can one of the conductors be neutralised. Let's look at a very simple example.







simulate this circuit &ndash; Schematic created using CircuitLab

Figure 1. (a) A floating battery. (b) An earthed battery.

In (a) the battery is floating. There is no ground connection unless there is a fault and one of the wires touches something earthed. Then the other-wire becomes live.

In (b) the battery negative has been connected to earth. It is grounded or neutralised and the other wire is 9 V with respect to earth.

One of the advantages of neutralising is that no fuses are required in the neutral line as there is no significant voltage with respect to earth.