It seems like the majority of people have an aversion to proving themselves to other people. I often hear comments like “I know I am (fill in the blank) and I shouldn’t have to prove it!” However, I often hear the same people wanting OTHERS to prove themelves. Why the double standard?
For those people who do this, it’s a social status thing. No one has to prove themselves to their (perceived) inferiors. I don’t think it’s majority behavior, though.
I think that’s just a rationalization for not living up to one’s own expectations. I have to prove myself every day on my job, because my responsibilities demand my very best effort all day, every day. And I’m a greedy bastard and I want a fat raise. Maybe even a mid-year bonus. 
I could explain that but then I don’t feel the need to prove to some anonymous stranger on the internet that I have the answer to their question.
It might help this discussion for the OP to clarify what sort of thing he meant by ‘proving yourself’, as in:
“I know that I’m a good sales man and I shouldn’t have to prove it.”
“I know that I’m honest and perceptive and I shouldn’t have to prove it.”
“I know that I’m Jesse Whitson of Fairview, California and I shouldn’t have to prove it.”
Etcetera, etcetera. All of them fit the general pattern you’ve described, but the reasons for people saying them would probably vary.
For me, the desire to “prove oneself” stems from an over-dependence on the opinions of others. To me, that type of thing always rings false. It always stikes me they’re trying to prove it to themselves as much as others-- that they are troubled by self-doubt.
I know who I am, and what my character is. Given time, others will know it as well. I don’t need to go to extraordinary lengths to demonstrate it.
Mostly as in your second quotation. As for the first, it’s pretty much a given that you must prove yourself at work, no matter how much pride you have. But, it seems when it comes to interpersonal relationships, people are more hesitant to prove that they are a good friend/partner/relative with good qualities. Yet, many of these people want other people to prove themselves, in my observation.
I don’t really have that many problems with proving myself. I personally find this to be quite effective strategy in “winning friends and influencing people”. I understand that most people are quite cynical and untrusting of others, and I don’t take it personally when others doubt me (if they don’t know me that well). Because I think proving yourself in both personal and professional situations can be effective, I wonder why many people avoid doing so.