I think you just have to honestly want equality and have general empathy for it to start to make sense. I'm a college graduate but it was in a technology field, and I never took an actual class on gender or race or any of that. So I don't really think my schooling contributed to the understanding of my own privilege and how it's associated to race.
Edit: And in the case of white people, we need to be able to accept that criticism is not an implicit attack against us. Too many white people get offended and instantly shutdown when you try to explain privilege because they want to believe they made it to where they were purely on their own merits free of an institutional advantage. That's not how the world works. We're all affected by the actions of others and the structure that community creates.
I think there are certainly qualities like empathy and thoughtfulness that help individual people understand their own privilege or concepts like structural racism but I feel that without genuine immersion in education specific to race, equity, identity, etc. whether that is in an academic setting or not.
I say this partly because I have met so many people who are traditionally liberal/progressive and believe themselves to be great allies to POC (as one example) and not racist at all but when you scratch beneath the surface they have tons of unresolved prejudice and racist beliefs. Generally I discover just how little they know about race and equity compared to how much they think they know. I truly believe that if one intends to be an effective ally, you need more than empathy, you have to know your shit. Caring is just step 1.