They generally want to encourage cis people to put their pronouns in their twitter bios because it helps to normalise the sharing of pronouns. Otherwise the only people who do it will be trans people, which leaves them singled out and immediately identifiable as a trans person.
In this case they hoped Carano would do it because 1) it's a positive gesture that costs her absolutely nothing, 2) it helps spread the idea among the huge, multi-generational fanbase who follow her and 3) it helps the plethora of trans people within that fanbase feel respected, included and welcome, as they so often aren't made to feel.
So in terms of "how is it their business?" and "why do they care?", the answer is "because they care about trans people and want to encourage people to do things that help make their very difficult lives slightly less difficult"