The singular they is not a new part of the English language. It’s been in use for over 600 years.
https://en.wikipedia.org/wiki/Singular_they
It’s just that you’re not taught all the grammatical forms so people don’t know how other people use the language. In the case of your example… I’d say “what are they doing here” instead of “what is they doing here”.
When somebody asks to be referred to by they/them instead of she/her or he/him there could be a bunch of reasons. Sometimes it’s because the person identifies as non-binary and doesn’t feel they are either a man or a woman. Another reason folks use they/them is because when you gender somebody it brings on a ton of expectations and you’re treated differently. So in an attempt to bring about a more egalitarian society they want to drop the use of genedered pronouns in most general speech.
I like the idea, but there’s not much evidence that languages without gendered pronouns have more gender equality than those who do gender their pronouns.
And most people who use gender neutral pronouns aren’t bothered when people make mistakes. What upsets people is when a speaker intentionally refuses to acknowledge their preference and invalidates them. Saying they are illegitimate.