Why is "he/him" "she/her" "they/them" such a big deal?

It's not always possible to tell someone's pronouns just looking at them. One of the simplest ways to respect someone's gender identification is to ask them for their pronouns and use them appropriately.

The incorrect use of pronouns can cause someone to feel insulted, invalidated, rejected, alienated, or dysphoric (often all of the above). The use of inclusive language by LGBTQ+ kids and adults significantly reduces experiences of melancholy, social anxiety, suicide ideation, and other detrimental mental health aspects, according to all major professional American psychological and psychiatric societies.

To not have to worry about which pronoun will be used for you based on how others perceive your gender is a privilege.