There's actually some interesting psychological research that suggests people primarily evaluate based on how you present things: kids are only cautious when their parents seem worried, and will be much calmer and more accepting if the parents act like something is no big deal. If you present a request casually, it's more likely to be casually accepted without thought. If you seem extremely anxious, people will pick up on that and get anxious themselves. Definitely a skill I have benefited from learning.

A sub-skill I would suggest is being okay with "no". I've found that if I ask for a big favor, get a "no", and just smile and move on, then people feel safer about me in the future - I didn't make them feel bad, so they don't have to be defensive about my future requests. It also makes it much easier for me to ask for the favor, and to come off casually, because I don't have any particular investment in a "yes" answer.