Remus Lupin fought alongside Frank and Alice Longbottom in the Order of the Phoenix. He probably knew their fate, how they were tortured into madness and forced to leave their son in the care of his grandmother.
Lupin probably guessed that Neville’s biggest fear would be similar to what he guessed Harry’s would be- Lord Voldemort, or in Neville’s case the death eater who tortured his parents - and was planning on stepping in before Neville faced the boggart so a bunch of 13 year olds wouldn’t see Bellatrix Lestrange cackling and spitting curses in their classroom.
But then Severus Snape insults Neville in front of his class, tries to shame a 13 year old boy in front of a new teacher, and Lupin is so angry. Because he had hoped that after 13 years, now they were adults, maybe Snape had gotten over himself a little and realized that he was a teacher and responsible for these kids, but apparently not. Apparently Snape was still the same petty, angry ,bitter child that he’d been in school, except now he’s bullying children half his age instead of yelling slurs at classmates.
And so Lupin sees the terror in Neville’s face and realizes that, oh my god he fears his potions teacher more than the people responsible for his parents. And Lupin is so indignant and outraged that he KNOWS he has to give this kid some leverage, some way to stand up to his tormentor, and gives that to him. Lupin stands up and tells Neville EXACTLY how to conquer his fear and, moreover, how to do it in front of all his classmates who laughed at him for being scared.
Remus Lupin starting off the school year by letting his students confront their biggest fears, so nothing else they face that year will be as scary. If you can beat your biggest fear, how bad can a test be? Or a big quidditch match? Or being rejected by the girl you ask out?
Remus Lupin truly, deeply, cared about his students. And Snape got him fired.
