Pronoun use is pretty straightforward. If someone says their pronouns, you use their pronouns to refer to them, just like you would anyone else. It may feel a little disorienting if you don't talk to many trans people and, for example, they use singular "they" or have a pronoun that seems different than the one that you would naturally assign to them based on their appearance, but it's not difficult and you can get used to it pretty quickly.
The reason that people are so sensitive and strict about this is because the stakes for trans people are very high -- some people don't believe trans people exist, should exist, or should have the same rights as cis people. Refusing to use the right pronouns reveals either a benign misunderstanding about trans people or a willful hostility towards their existence. The latter is extremely common and can be both hurtful and often scary, as trans people, especially trans people of color, are often subject to violence because of their identities.