Just like calling someone by the right name, using someone’s pronouns is is an important way of signalling respect, courtesy and acceptance. When we aren’t addressed correctly, many of us will feel unseen. This is particularly true for someone who has worked hard to be their true self. Ultimately, asking someone what their pronouns are is really about asking someone what the respectful way to refer to them is (if you are not using their name.)
Misgendering someone (i.e. referring to someone using a form of address or pronoun that doesn't correspond with their gender identity) can feel like disrespect and can be both damaging and hurtful for someone who has worked hard to come out and be themselves. A person's identity is important and when people use the correct pronouns to identify someone, that person feels like they are being themselves - and being understood as that person.
People think carefully about the pronouns they want to use and why.