Working of a Doubly-Fed Induction generator (DFIG) in wind turbines
I am unable to understand how a Doubly-Fed Induction generator works
  (in Wind Energy conversion system)?


I think the document you linked is a little confusing and I would recommend this better document instead. Entitled "Principles of Doubly-Fed Induction Generators (DFIG)" and produced by Lab-Volt.

However, in simple terms, think about a regular 3 phase synchronous generator first; DC is applied to the rotor, the shaft is rotated at synchronous speed and the generator produces an AC output of the correct frequency (50 or 60 Hz).

However, if the shaft isn't rotated at synchronous speed but a somewhat slower speed (as what usually happens in a wind generator), you can still produce a synchronous output frequency (50 or 60 Hz) by feeding the rotor with an AC current of "so many hertz" instead of DC. The rotor frequency now "makes up the difference" between shaft speed and synchronous speed: -



Picture source.

The important things to notice is that the rotor voltage and current is derived from an AC/DC converter (rectifier) followed by a DC/AC convertor. The DC/AC converter is an inverter whose frequency can be set to a value that  "makes up the difference" between shaft speed and synchronous speed.

Not shown in the diagram is that the control block needs to accurately measure shaft speed in order to calculate the rotor frequency required to produce a synchronous AC output from the stator.


  It would be helpful if somebody is able to explain the phenomenon with
  the help of a phasor diagram showing the voltage, current and the
  field directions.


If you understand the phasor diagrams for a standard synchronous generator and the relationship between slip and driving frequency for a 3 ph inductor motor, then it will be clear. If you don't understand these then you need to ask a more appropriate question.