Andy DS said:
the expansion of spacetime of the Universe doesn't seem to have any effect on the small scale ie the solar system.
That is true, but gravitational waves are not the same thing as the expansion of the universe. The expansion of the universe, in this context, is best viewed as providing a "background" spacetime geometry, on which gravitational waves are localized fluctuations.
Andy DS said:
if gravitational waves from merging black holes stretch space time which has let's say infinite elasticity
There is a viewpoint in which spacetime can be viewed as a sort of elastic medium (among others, Sakharov, the famous Russian physicist, worked on this sort of model in the 1960s), but in this view, the elasticity is not infinite.
Andy DS said:
how does it manage to stretch the earth
As has already been noted, LIGO does not measure the stretching and squeezing of the Earth by gravitational waves. It measures the stretching and squeezing of the arms of a very precisely constructed and sensitive detector which, as has been noted, is carefully isolated from all other external sources of vibration, including the Earth.
There is another type of gravitational wave detector called a bar detector, originally invented by Weber and worked on several decades ago by a Russian team led by Braginsky (this work is described in the popular book
Black Holes and Time Warps by Kip Thorne), which does measure the stretching and squeezing of a solid object (a large bar, usually of a metal like iron) by gravitational waves. Braginsky's work showed that this type of detector has serious limitations which probably make it unable to detect gravitational waves coming to Earth from distant sources, given the expected strength of those waves (now confirmed by LIGO). That limitation was one of the main reasons why LIGO used interferometer-type detectors.