Depends what you mean by side characters, I assume you mean the kind that are introduced in flashbacks or get moments here and there whilst not being too major.
With Corazon, he was attached to Law and his life, significance and death are linked to Law. He was an inspiration and role model for Law so no surprise people grew attached, myself included.
Same for Bellamy and a lot of characters that appear in backstories, their lives and sometimes deaths hold so much weight because of the impact it had on already existing characters. We see her efforts to take care of Nami and Nojiko, how she believes even if she isn't their biological parent, she still loves them and we can see that in action with her sacrifice.
As for people enjoying them overall and appreciating them even if they have small roles, probably just the characters being likable. Like in Alabasta, Toto (Koza, Vivi's childhood friend's father) was determined to continue digging for the sake of finding water despite all the sandstorms Crocodile had caused and even if his moments with Luffy are simple, they're pretty effective in portraying the resilience of some of the people of Alabsta.
Plus Oda still tries when it comes to certain characters despite their small roles. It's all about the effectiveness of a backstory or the presence a character has even if they aren't around for too long.
Overall, a character doesn't need to be on screen for too long to leave a mark, especially when they're tied to an already established character. Not to mention, appreciating side characters doesn't always mean you're crying over them, sometimes a story can be touching and important in regards to building up to larger themes while simultaneously standing on its own and that's enough.