How and when are the relativistic corrections applied to GPS satellites? It is known that there is a need to correct the onboard clocks to reduce the time difference from 38μs to 50ns. Where is relativity playing its role here? Why cant the clocks be simply synchronised with the ground clocks through telecommunication? If these are not possible then how are the clocks corrected? 
 A: They are synchronized through communication.  It's just that you'd rather the corrections be as small as possible.  The clock on board should "tick" closely to the time you want.  The ability to modify the rate of the clock is limited, so only small rate changes can be accommodated.  So the corrections are built in to the rate that the onboard clock counts.
When the first test vehicle was made, a frequency synthesizer was included that could apply additional corrections should they be needed (in other words, if the GR corrections were wrong or unnecessary).  The time signals were found to be correct and the synthesizer was never energized.  Later vehicles did not include any provisions to modify the frequency in this way.
Some background here, but I have been told that contrary to the details in that paper, the vehicle was never operated with the synthesizer on because the time signals without it running were in spec.  Either way, that's just an operational detail of the first vehicle.  I've omitted further references that appear in that paper.

There is an interesting story about this frequency offset. At the time of launch of the NTS-2
  satellite (23 June 1977), which contained the first Cesium atomic clock to be placed in orbit, it was
  recognized that orbiting clocks would require a relativistic correction, but there was uncertainty
  as to its magnitude as well as its sign. Indeed, there were some who doubted that relativistic
  effects were truths that would need to be incorporated! A frequency synthesizer was built into
  the satellite clock system so that after launch, if in fact the rate of the clock in its final orbit was
  that predicted by general relativity, then the synthesizer could be turned on, bringing the clock
  to the coordinate rate necessary for operation. After the Cesium clock was turned on in NTS-2,
  it was operated for about 20 days to measure its clock rate before turning on the synthesizer.
  The frequency measured during that interval was $+442.5$ parts in $10^{12}$ compared to clocks on the
  ground, while general relativity predicted $+446.5$ parts in $10^{12}$. The difference was well within the
  accuracy capabilities of the orbiting clock. This then gave about a $1\%$ verification of the combined
  second-order Doppler and gravitational frequency shift effects for a clock at 4.2 earth radii.

A: 
Where is relativity playing its role here?

This part is well explained in Wikipedia: Effects of relativity on GPS

Why cant the clocks be simply synchronised with the ground clocks
  through telecommunication?

They can, and in fact that's exactly part of what the GPS Control Segment does: measuring the satellite clock errors; except that the GPS satellite clocks are not forced to tick in synchrony; instead, the amount of their error is broadcast to users so that it can be subtracted as a correction in software calculations.
If the onboard relativistic hardware correction didn't exist, the satellite clock error would just be larger; the correction accounts for the bulk of the expected deviation between nominal carrier frequency and what is actually received on the ground, as per GPS ICD:

"The carrier frequencies for the L1 and L2 signals shall be coherently
  derived from a common frequency source within the SV [satellite
  vehicle]. The nominal frequency of this source -- as it appears to an
  observer on the ground -- is 10.23 MHz. The SV carrier frequency and
  clock rates -- as they would appear to an observer located in the SV
  -- are offset to compensate for relativistic effects. The clock rates are offset by ∆f/f = -4.4647E-10, equivalent to a change in the P-code
  chipping rate of 10.23 MHz offset by a ∆ f = -4.5674E-3 Hz. This is
  equal to 10.2299999954326 MHz. The nominal carrier frequencies (f0)
  shall be 1575.42 MHz, and 1227.6 MHz for L1 and L2, respectively."

A: Due to relativity, the clocks on the GPS satellites move fast by about 38 µs per day. Which would be a problem, but not that big a problem because they all move fast by the same amount. Still you'd need to synchronize the clocks from time to time, because the satellite's position in space also depends on the clock. 
HOWEVER, if you do that once a week, compensating for about 270 µs, you'd have to be able to do that absolutely at the same time for all satellites. If I'm driving happily through a town, directed by my TomTom, and the first satellite changes by 270 µs = 80 km, and then the second satellite changes ten seconds later, and so on, my TomTom will have a major problem with that, and so will I. 
Much better to make sure that the clocks compensate for the known 38 µs per day by themselves all the time, so the compensation is only for a few nanoseconds. 
