I'd say arrogance is expecting success without actual certainty that it will occur. Some people are good enough at what they do that their odds of success are high enough to make them seem trustworthy when they speak confidently.

Others have no real reason to expect to be successful, having never experienced success in what they are attempting.

So to me, the best way to avoid arrogance is to take an honest assessment of your experience and whether you have actually experienced success in the arena you are attempting to enter, and if you can't point to past successes, don't speak as if you expect to succeed this time.

This is something that comes with age, I believe. As you get older you gain more experience, so you understand that success is difficult to achieve. This does not necessarily mean approaching things with a defeatist attitude, but rather to take a more comprehensive look at the challenges and not merely rely on your own unproven abilities and proceed accordingly.