The Mandalorian is, in large part, a character study of a very straightforward character. The central hero Din Djarin is a proud member of his warrior culture who finds himself charged with the protection of a powerful young foundling. Since the show is named after Djarin, it gets fairly deep into the masked man's psyche. One of his most notable traits is his dislike and distrust of droids.
Droids have an interesting role in the Star Wars universe. They've been henchmen, heroes, villains minor side characters, and workers across the entire franchise. They're rarely the main characters, but they're always around. Curiously, the series never really plays with themes of AI or machine labor, but there are quite a few characters who don't like droids.
Why doesn't Din Djarin trust Droids?
Throughout The Mandalorian, fans see Din Djarin meet a similar situation multiple times. He needs someone to do a job for him. Maybe he wants his ship repaired, or some new weaponry or someone to watch Grogu. One or more droids are around, so they offer their services. Din refuses, often adding extra time, cost, or effort to the project. People are generally willing to leave the machines out of it, though they often seem a bit unhappy about it. He even has the droid removed from his new ship. Djarin doesn't want to be around droids, he doesn't even trust them to do their jobs. It may seem irrational, but his hate and fear of machines is understandable to those who know where Djarin comes from.
As fans know, Din Djarin wasn't born into the Mandalorian culture. He was a foundling that was raised to be a Child of the Watch from a young age. Born on Aq Vetina, Din Djarin's parents were killed in the Fall of the Republic which led to the creation of the Empire. His entire planet's population was slaughtered and he faced death, only to be saved by a Mandalorian. In the early era of the Empire, they still used battle droids for all of their military engagements. That means that Din Djarin's most formative early memory is that of everyone he's ever known being gunned down by droids. He can't stand machine life in general because, on some level, he still sees them as the foot soldiers that took his parents away. He still remembers looking down the barrel of a super battle droid's gun at the moment before his savior gave him his new life. Though this trauma has created a lifelong disdain for droids, he does appear to be getting over it.
Is Din Djarin changing his ways?
As The Mandalorian goes on, Din Djarin is arguably moving past some of his worst traits. He's taciturn, cold, and confrontational when he first appears. Gradually, he opened his heart to others and assembled a cast of friends. Din Djarin has softened to his adopted foundling, but he's also become warmer to those around him. In the first episodes, his relationships with people who aren't fellow Children of the Watch are distant and transactional. He has managed to upgrade some of his allies to friends. One of the most notable moments of change for the character came with his relationship with IG-11.
IG-11 is an assassin droid who takes bounty-hunting gigs from the same guild that previously employed Djarin. Mando encounters him on the job, joining the machine on his quest to eliminate an ominous "asset." He willingly fights alongside a droid, but their initial interaction is practical. Djarin blows the bot's brains out when the mission is finished, choosing to save The Child from his fate. When Djarin next meets IG-11, the machine has been reprogrammed to act as a protector. The Mandalorian and the droid fight back-to-back to escape a bad situation. Djarin removes his helmet in front of IG-11, a cardinal sin among his people, to allow the machine to treat his wounds. IG-11 sacrifices himself to save Grogu, noting that Djarin is genuinely sad to see him go. Now, in the third season premiere, he's trying to get the only droid he's ever cared for back into action. That's a great deal of character growth in one relationship.
Din Djarin has gone from rejecting a ride to his ship from a vehicle with a droid to going out of his way to get one repaired. His childhood experience taught him that droids could be used to do incredible harm across the galaxy, but instead of blaming the people who programmed them, he blamed the machines. Now, on Mando's long journey of personal growth, he's learning to judge each droid as an individual. Of course, he still doesn't care for most individuals, but at least he's stopped writing off entire types of sentient beings. It's a baby step, but it's a step.