How can a DFIG have total control of reactive and active power?
In order for an INDUCTION generator to function (at a basic level), the rotor must be excited, i.e. flux is created, then driven faster than the stator frequency, which is connected to and determined by the grid. So because an induction machine needs AC to create that rotor flux, that AC must come FROM the grid. The prime mover, in this case the wind turbine, then drives the rotor to be faster than the stator, hence the rotor frequency is greater than the stator frequency (negative slip) so the stator pumps energy into the grid. But if the wind speed ever drops, the rotor speed becomes less than the stator frequency and the machine becomes a motor, not a generator (actually, it is shut down to prevent that). 

What's different about a Doubly Fed Induction Generator (DFIG) is that the control system, through the power electronic converters shown, is able to keep the rotor frequency higher than the stator frequency regardless of what the ACTUAL rotor speed is; it's essentially a Variable Frequency Drive applied to the rotor. That rotor flux current still has to come from the grid, but at the same time any reactive current, which takes place out of phase with the flux current, can be recovered and also sent back into the grid, increasing the efficiency of the generator as well as allowing it to function in a wider range of wind speeds. Hence the arrows showing current flowing both ways.