During the detection of gravitational waves by LIGO machine, why wasn't the laser light used in the detection, also stretched as a result of the stretching of space time fabric?
Yes, it stretches and squeezes the light, i.e., the wavelength, and does tthe opposite (lowers and raises) the frequency, and keeps the speed of light the same. What the interferometer sees is that the light in the longer path (because that path was stretched) had to travel further, a peak took longer to arrive (in the stretch portion, the opposite in the squeeze).
So, say if a second Ray of light which did not see that stretching (keep to stretching, we mean alternatively stretching and sneezing in concert with the gravitational wave), and at some instant say it arrived at a peak, then the first one would not be at a peak, there would be a phase difference. The interferometer is sensitive to that phase difference. If that first one was at a trough, it'd cancel out with the second Ray.
The fact that the light going through the gravitational wave arrives a little later allows us to know that the gravitational wave was there. The wavelength is indeed stretched (and later squeezed), but the speed of light is the same, and so it arrives a little later (for a given phase)
In fact, of course, LIGO uses perpendicular legs, and when one is stretched the other one is squeezed, and it changes, so we see the changing phase difference as an electrical wave on the output.
This is answered and described in LIGOs FAQ at https://www.ligo.caltech.edu/page/faq