Metallicity relates to the quantity of elements in an object that are heavier than hydrogen and helium. I often see this terminology in recent studies of gravitational waves such as this paper.
If I want to study the formation of stars or any other stellar remnants such as (neutron star black hole-NSBH, binary neutron star-BNS), what information do we get from studying metallicity specifically?
What precisely does metallicity include in astronomy, and why is it necessary to study it?
As you say, metallicity in this context refers to the proportion of the interstellar medium a star forms from that consists of elements heavier than helium. It can be expressed as a (mass) fraction (usually called $Z$) or sometimes as a (logarithmic) ratio with respect to the metallicity inferred for the Sun ([M/H] or sometimes [Fe/H]).
Metallicity plays an important role in several ways with regard to neutron stars, black holes and the binary systems they form. Most of these are connected with the effects the composition of a gas has on its opacity to radiation - generally speaking, high metallicity leads to higher opacity.
Opacity may control the mass spectrum of stars that form. In particular, it is expected that low-metallicity gas might be capable of making a bigger proportion of the high-mass stars that end their lives as neutron stars and black holes.
Opacity can heavily influence the amount of mass that is lost from high-mass stars. That is because they have radiatively driven winds and high opacity leads to greater mass loss. The amount of mass a star possesses as it heads towards the end of its life may control what its ultimate fate may be; whether or what kind of supernovae will result; and what kind and mass of remnant will be left behind, and so all this is expected to be metallicity-dependent.
Opacity has an effect on the structure of high-mass stars, as well as their mass loss, and both of these are important factors in assessing how binary systems involving high-mass stars evolve. The merging binaries observed to produce gravitational waves must have components that had previously been brought quite close together by stellar/binary evolution in order for gravitational wave losses to subsequently become important.
Since the metallicity of gas in the universe is gradually being enriched from the time of first star formation onwards and most strongly in actively star-forming galaxies, then the influence of metallicity on stellar and binary evolution and the production of gravitational wave progenitors is intimately wrapped up with when and where in the universe the gravitational wave sources are located.
