Earlier quoted context omitted.
I have a friend from the western US who was explicitly taught by her (white) mother that you always refuse a favor the first time it is offered. There were many months of me never doing any favors for her before we figured that one out...
I'm from a western state and my family is pretty white. I've never heard this. You definitely don't glob onto any/all favors and you shouldn't accept something you wouldn't be willing to reciprocate in the same position, else you're bound to be seen as social baggage eventually. But if you need something and someone offers: sure, take it. There's no social dance to it. Just don't be a leech, but accept help when it's…
There is. What constitutes a leech varies from person to person, from culture to culture. Accepting help could easily lead to resentment, it's entirely possible they were offering help to seem generous while simultaneously expecting to be refused. Offering help at all could be offensive because you're in a position of strength while they're in a position of weakness, it implies they need you, ingratiates them with you, puts them in your debt.
Correctly navigating these waters requires instant judgements based on huge amounts of social information like status, reputation, personality, context, non-verbal cues like tone of voice and body language. It is difficult to do this deliberately because during conversations there is not enough time to deeply analyze anything. It's best left to an uncounscious mind honed sharp by repeated practice.